The Fourier Transform and applications - Mihalis Kolountzakis
Transcript of The Fourier Transform and applications - Mihalis Kolountzakis
![Page 1: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/1.jpg)
The Fourier Transform and applications
Mihalis Kolountzakis
University of Crete
January 2006
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 1 / 36
![Page 2: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/2.jpg)
Groups and Haar measure
Locally compact abelian groups:
Integers Z = {. . . ,−2,−1, 0, 1, 2, . . .}Finite cyclic group Zm = {0, 1, . . . ,m − 1}: addition modm
Reals RTorus T = R/Z: addition of reals mod1
Products: Zd , Rd , T× R, etc
Haar measure on G = translation invariant on G : µ(A) = µ(A + t).Unique up to scalar multiple.
Counting measure on ZCounting measure on Zm, normalized to total measure 1 (usually)
Lebesgue measure on RLebesgue masure on T viewed as a circle
Product of Haar measures on the components
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 2 / 36
![Page 3: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/3.jpg)
Groups and Haar measure
Locally compact abelian groups:
Integers Z = {. . . ,−2,−1, 0, 1, 2, . . .}
Finite cyclic group Zm = {0, 1, . . . ,m − 1}: addition modm
Reals RTorus T = R/Z: addition of reals mod1
Products: Zd , Rd , T× R, etc
Haar measure on G = translation invariant on G : µ(A) = µ(A + t).Unique up to scalar multiple.
Counting measure on Z
Counting measure on Zm, normalized to total measure 1 (usually)
Lebesgue measure on RLebesgue masure on T viewed as a circle
Product of Haar measures on the components
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 2 / 36
![Page 4: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/4.jpg)
Groups and Haar measure
Locally compact abelian groups:
Integers Z = {. . . ,−2,−1, 0, 1, 2, . . .}Finite cyclic group Zm = {0, 1, . . . ,m − 1}: addition modm
Reals RTorus T = R/Z: addition of reals mod1
Products: Zd , Rd , T× R, etc
Haar measure on G = translation invariant on G : µ(A) = µ(A + t).Unique up to scalar multiple.
Counting measure on ZCounting measure on Zm, normalized to total measure 1 (usually)
Lebesgue measure on RLebesgue masure on T viewed as a circle
Product of Haar measures on the components
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 2 / 36
![Page 5: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/5.jpg)
Groups and Haar measure
Locally compact abelian groups:
Integers Z = {. . . ,−2,−1, 0, 1, 2, . . .}Finite cyclic group Zm = {0, 1, . . . ,m − 1}: addition modm
Reals R
Torus T = R/Z: addition of reals mod1
Products: Zd , Rd , T× R, etc
Haar measure on G = translation invariant on G : µ(A) = µ(A + t).Unique up to scalar multiple.
Counting measure on ZCounting measure on Zm, normalized to total measure 1 (usually)
Lebesgue measure on R
Lebesgue masure on T viewed as a circle
Product of Haar measures on the components
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 2 / 36
![Page 6: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/6.jpg)
Groups and Haar measure
Locally compact abelian groups:
Integers Z = {. . . ,−2,−1, 0, 1, 2, . . .}Finite cyclic group Zm = {0, 1, . . . ,m − 1}: addition modm
Reals RTorus T = R/Z: addition of reals mod1
Products: Zd , Rd , T× R, etc
Haar measure on G = translation invariant on G : µ(A) = µ(A + t).Unique up to scalar multiple.
Counting measure on ZCounting measure on Zm, normalized to total measure 1 (usually)
Lebesgue measure on RLebesgue masure on T viewed as a circle
Product of Haar measures on the components
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 2 / 36
![Page 7: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/7.jpg)
Groups and Haar measure
Locally compact abelian groups:
Integers Z = {. . . ,−2,−1, 0, 1, 2, . . .}Finite cyclic group Zm = {0, 1, . . . ,m − 1}: addition modm
Reals RTorus T = R/Z: addition of reals mod1
Products: Zd , Rd , T× R, etc
Haar measure on G = translation invariant on G : µ(A) = µ(A + t).Unique up to scalar multiple.
Counting measure on ZCounting measure on Zm, normalized to total measure 1 (usually)
Lebesgue measure on RLebesgue masure on T viewed as a circle
Product of Haar measures on the components
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 2 / 36
![Page 8: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/8.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.
χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ TG = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ ZG = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ RG = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 9: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/9.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ TG = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ ZG = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ RG = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 10: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/10.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ TG = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ ZG = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ RG = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 11: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/11.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ TG = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ ZG = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ RG = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 12: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/12.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ T
G = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ ZG = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ RG = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 13: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/13.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ TG = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ Z
G = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ RG = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 14: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/14.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ TG = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ ZG = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ R
G = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 15: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/15.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ TG = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ ZG = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ RG = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 16: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/16.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ TG = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ ZG = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ RG = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 17: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/17.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ TG = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ ZG = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ RG = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 18: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/18.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ TG = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ ZG = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ RG = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 19: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/19.jpg)
Characters and the dual group
Character is a (continuous) group homomorphism from G to themultiplicative group U = {z ∈ C : |z | = 1}.χ : G → U satsifies χ(h + g) = χ(h)χ(g)
If χ, ψ are characters then so is χψ (pointwise product). Write χ+ ψfrom now on instead of χψ.
Group of characters (written additively) G is the dual group of G
G = Z =⇒ G = T: the functions χx(n) = exp(2πixn), x ∈ TG = T =⇒ G = Z: the functions χn(x) = exp(2πinx), n ∈ ZG = R =⇒ G = R: the functions χt(x) = exp(2πitx), t ∈ RG = Zm =⇒ G = Zm: the functions χk(n) = exp(2πikn/m), k ∈ Zm
G = A× B =⇒ G = A× B
Example: G = T× R =⇒ G = Z× R. The characters areχn,t(x , y) = exp(2πi(nx + ty)).
G is compact ⇐⇒ G is discrete
Pontryagin duality:G = G .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 3 / 36
![Page 20: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/20.jpg)
The Fourier Transform of integrable functions
f ∈ L1(G ). That is ‖f ‖1 :=∫G |f (x)| dµ(x) <∞
If G is finite then L1(G ) is all functions G → CThe FT of f is f : G → C defined by
f (χ) =
∫G
f (x)χ(x) dµ(x), χ ∈ G
Example: G = T (“Fourier coefficients”):
f (n) =
∫T
f (x)e−2πinx dx , n ∈ Z
Example: G = R (“Fourier transform”):
f (ξ) =
∫T
f (x)e−2πiξx dx , ξ ∈ R
Example: G = Zm (“Discrete Fourier transform or DFT”):
f (k) =1
m
m−1∑j=0
f (j)e−2πikj/m, k ∈ Zm
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 4 / 36
![Page 21: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/21.jpg)
The Fourier Transform of integrable functions
f ∈ L1(G ). That is ‖f ‖1 :=∫G |f (x)| dµ(x) <∞
If G is finite then L1(G ) is all functions G → C
The FT of f is f : G → C defined by
f (χ) =
∫G
f (x)χ(x) dµ(x), χ ∈ G
Example: G = T (“Fourier coefficients”):
f (n) =
∫T
f (x)e−2πinx dx , n ∈ Z
Example: G = R (“Fourier transform”):
f (ξ) =
∫T
f (x)e−2πiξx dx , ξ ∈ R
Example: G = Zm (“Discrete Fourier transform or DFT”):
f (k) =1
m
m−1∑j=0
f (j)e−2πikj/m, k ∈ Zm
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 4 / 36
![Page 22: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/22.jpg)
The Fourier Transform of integrable functions
f ∈ L1(G ). That is ‖f ‖1 :=∫G |f (x)| dµ(x) <∞
If G is finite then L1(G ) is all functions G → CThe FT of f is f : G → C defined by
f (χ) =
∫G
f (x)χ(x) dµ(x), χ ∈ G
Example: G = T (“Fourier coefficients”):
f (n) =
∫T
f (x)e−2πinx dx , n ∈ Z
Example: G = R (“Fourier transform”):
f (ξ) =
∫T
f (x)e−2πiξx dx , ξ ∈ R
Example: G = Zm (“Discrete Fourier transform or DFT”):
f (k) =1
m
m−1∑j=0
f (j)e−2πikj/m, k ∈ Zm
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 4 / 36
![Page 23: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/23.jpg)
The Fourier Transform of integrable functions
f ∈ L1(G ). That is ‖f ‖1 :=∫G |f (x)| dµ(x) <∞
If G is finite then L1(G ) is all functions G → CThe FT of f is f : G → C defined by
f (χ) =
∫G
f (x)χ(x) dµ(x), χ ∈ G
Example: G = T (“Fourier coefficients”):
f (n) =
∫T
f (x)e−2πinx dx , n ∈ Z
Example: G = R (“Fourier transform”):
f (ξ) =
∫T
f (x)e−2πiξx dx , ξ ∈ R
Example: G = Zm (“Discrete Fourier transform or DFT”):
f (k) =1
m
m−1∑j=0
f (j)e−2πikj/m, k ∈ Zm
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 4 / 36
![Page 24: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/24.jpg)
The Fourier Transform of integrable functions
f ∈ L1(G ). That is ‖f ‖1 :=∫G |f (x)| dµ(x) <∞
If G is finite then L1(G ) is all functions G → CThe FT of f is f : G → C defined by
f (χ) =
∫G
f (x)χ(x) dµ(x), χ ∈ G
Example: G = T (“Fourier coefficients”):
f (n) =
∫T
f (x)e−2πinx dx , n ∈ Z
Example: G = R (“Fourier transform”):
f (ξ) =
∫T
f (x)e−2πiξx dx , ξ ∈ R
Example: G = Zm (“Discrete Fourier transform or DFT”):
f (k) =1
m
m−1∑j=0
f (j)e−2πikj/m, k ∈ Zm
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 4 / 36
![Page 25: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/25.jpg)
The Fourier Transform of integrable functions
f ∈ L1(G ). That is ‖f ‖1 :=∫G |f (x)| dµ(x) <∞
If G is finite then L1(G ) is all functions G → CThe FT of f is f : G → C defined by
f (χ) =
∫G
f (x)χ(x) dµ(x), χ ∈ G
Example: G = T (“Fourier coefficients”):
f (n) =
∫T
f (x)e−2πinx dx , n ∈ Z
Example: G = R (“Fourier transform”):
f (ξ) =
∫T
f (x)e−2πiξx dx , ξ ∈ R
Example: G = Zm (“Discrete Fourier transform or DFT”):
f (k) =1
m
m−1∑j=0
f (j)e−2πikj/m, k ∈ Zm
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 4 / 36
![Page 26: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/26.jpg)
Elementary properties of the Fourier Transform
Linearity: λf + µg = λf + µg .
Symmetry: f (−x) = f (x), f (x) = f (−x)
Real f : then f (x) = f (−x)
Translation: if τ ∈ G , ξ ∈ G , fτ (x) = f (x − τ) thenfτ (ξ) = ξ(τ) · f (ξ).
Example: G = T: f (x − θ)(n) = e−2πinθ f (n), for θ ∈ T, n ∈ Z.
Modulation: If χ, ξ ∈ G then χ(x)f (x)(ξ) = f (ξ − χ).
Example: G = R: e2πitx f (x)(ξ) = f (ξ − t).
f , g ∈ L1(G ): their convolution is f ∗ g(x) =∫G f (t)g(x − t) dµ(t).
Then ‖f ∗ g‖1 ≤ ‖f ‖1‖g‖1 and
f ∗ g(ξ) = f (ξ) · g(ξ), ξ ∈ G
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 5 / 36
![Page 27: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/27.jpg)
Elementary properties of the Fourier Transform
Linearity: λf + µg = λf + µg .
Symmetry: f (−x) = f (x), f (x) = f (−x)
Real f : then f (x) = f (−x)
Translation: if τ ∈ G , ξ ∈ G , fτ (x) = f (x − τ) thenfτ (ξ) = ξ(τ) · f (ξ).
Example: G = T: f (x − θ)(n) = e−2πinθ f (n), for θ ∈ T, n ∈ Z.
Modulation: If χ, ξ ∈ G then χ(x)f (x)(ξ) = f (ξ − χ).
Example: G = R: e2πitx f (x)(ξ) = f (ξ − t).
f , g ∈ L1(G ): their convolution is f ∗ g(x) =∫G f (t)g(x − t) dµ(t).
Then ‖f ∗ g‖1 ≤ ‖f ‖1‖g‖1 and
f ∗ g(ξ) = f (ξ) · g(ξ), ξ ∈ G
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 5 / 36
![Page 28: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/28.jpg)
Elementary properties of the Fourier Transform
Linearity: λf + µg = λf + µg .
Symmetry: f (−x) = f (x), f (x) = f (−x)
Real f : then f (x) = f (−x)
Translation: if τ ∈ G , ξ ∈ G , fτ (x) = f (x − τ) thenfτ (ξ) = ξ(τ) · f (ξ).
Example: G = T: f (x − θ)(n) = e−2πinθ f (n), for θ ∈ T, n ∈ Z.
Modulation: If χ, ξ ∈ G then χ(x)f (x)(ξ) = f (ξ − χ).
Example: G = R: e2πitx f (x)(ξ) = f (ξ − t).
f , g ∈ L1(G ): their convolution is f ∗ g(x) =∫G f (t)g(x − t) dµ(t).
Then ‖f ∗ g‖1 ≤ ‖f ‖1‖g‖1 and
f ∗ g(ξ) = f (ξ) · g(ξ), ξ ∈ G
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 5 / 36
![Page 29: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/29.jpg)
Elementary properties of the Fourier Transform
Linearity: λf + µg = λf + µg .
Symmetry: f (−x) = f (x), f (x) = f (−x)
Real f : then f (x) = f (−x)
Translation: if τ ∈ G , ξ ∈ G , fτ (x) = f (x − τ) thenfτ (ξ) = ξ(τ) · f (ξ).
Example: G = T: f (x − θ)(n) = e−2πinθ f (n), for θ ∈ T, n ∈ Z.
Modulation: If χ, ξ ∈ G then χ(x)f (x)(ξ) = f (ξ − χ).
Example: G = R: e2πitx f (x)(ξ) = f (ξ − t).
f , g ∈ L1(G ): their convolution is f ∗ g(x) =∫G f (t)g(x − t) dµ(t).
Then ‖f ∗ g‖1 ≤ ‖f ‖1‖g‖1 and
f ∗ g(ξ) = f (ξ) · g(ξ), ξ ∈ G
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 5 / 36
![Page 30: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/30.jpg)
Elementary properties of the Fourier Transform
Linearity: λf + µg = λf + µg .
Symmetry: f (−x) = f (x), f (x) = f (−x)
Real f : then f (x) = f (−x)
Translation: if τ ∈ G , ξ ∈ G , fτ (x) = f (x − τ) thenfτ (ξ) = ξ(τ) · f (ξ).
Example: G = T: f (x − θ)(n) = e−2πinθ f (n), for θ ∈ T, n ∈ Z.
Modulation: If χ, ξ ∈ G then χ(x)f (x)(ξ) = f (ξ − χ).
Example: G = R: e2πitx f (x)(ξ) = f (ξ − t).
f , g ∈ L1(G ): their convolution is f ∗ g(x) =∫G f (t)g(x − t) dµ(t).
Then ‖f ∗ g‖1 ≤ ‖f ‖1‖g‖1 and
f ∗ g(ξ) = f (ξ) · g(ξ), ξ ∈ G
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 5 / 36
![Page 31: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/31.jpg)
Elementary properties of the Fourier Transform
Linearity: λf + µg = λf + µg .
Symmetry: f (−x) = f (x), f (x) = f (−x)
Real f : then f (x) = f (−x)
Translation: if τ ∈ G , ξ ∈ G , fτ (x) = f (x − τ) thenfτ (ξ) = ξ(τ) · f (ξ).
Example: G = T: f (x − θ)(n) = e−2πinθ f (n), for θ ∈ T, n ∈ Z.
Modulation: If χ, ξ ∈ G then χ(x)f (x)(ξ) = f (ξ − χ).
Example: G = R: e2πitx f (x)(ξ) = f (ξ − t).
f , g ∈ L1(G ): their convolution is f ∗ g(x) =∫G f (t)g(x − t) dµ(t).
Then ‖f ∗ g‖1 ≤ ‖f ‖1‖g‖1 and
f ∗ g(ξ) = f (ξ) · g(ξ), ξ ∈ G
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 5 / 36
![Page 32: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/32.jpg)
Orthogonality of characters on compact groups
If G is compact (=⇒ total Haar measure = 1) then characters are inL1(G ), being bounded.
If χ ∈ G then∫Gχ(x) dx =
∫Gχ(x + g) dx = χ(g)
∫Gχ(x) dx ,
so∫G χ = 0 if χ nontrivial, 1 if χ is trivial (= 1).
If χ, ψ ∈ G then χ(x)ψ(−x) is also a character. Hence
〈χ, ψ〉 =
∫Gχ(x)ψ(x) dx =
∫Gχ(x)ψ(−x) dx =
{1 χ = ψ0 χ 6= ψ
Fourier representation (inversion) in Zm: G = Zm =⇒ the m
characters form a complete orthonormal set in L2(G ):
f (x) =m−1∑k=0
〈f (·), e2πik·〉e2πikx =m−1∑k=0
f (k)e2πikx
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 6 / 36
![Page 33: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/33.jpg)
Orthogonality of characters on compact groups
If G is compact (=⇒ total Haar measure = 1) then characters are inL1(G ), being bounded.
If χ ∈ G then∫Gχ(x) dx =
∫Gχ(x + g) dx = χ(g)
∫Gχ(x) dx ,
so∫G χ = 0 if χ nontrivial, 1 if χ is trivial (= 1).
If χ, ψ ∈ G then χ(x)ψ(−x) is also a character. Hence
〈χ, ψ〉 =
∫Gχ(x)ψ(x) dx =
∫Gχ(x)ψ(−x) dx =
{1 χ = ψ0 χ 6= ψ
Fourier representation (inversion) in Zm: G = Zm =⇒ the m
characters form a complete orthonormal set in L2(G ):
f (x) =m−1∑k=0
〈f (·), e2πik·〉e2πikx =m−1∑k=0
f (k)e2πikx
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 6 / 36
![Page 34: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/34.jpg)
Orthogonality of characters on compact groups
If G is compact (=⇒ total Haar measure = 1) then characters are inL1(G ), being bounded.
If χ ∈ G then∫Gχ(x) dx =
∫Gχ(x + g) dx = χ(g)
∫Gχ(x) dx ,
so∫G χ = 0 if χ nontrivial, 1 if χ is trivial (= 1).
If χ, ψ ∈ G then χ(x)ψ(−x) is also a character. Hence
〈χ, ψ〉 =
∫Gχ(x)ψ(x) dx =
∫Gχ(x)ψ(−x) dx =
{1 χ = ψ0 χ 6= ψ
Fourier representation (inversion) in Zm: G = Zm =⇒ the m
characters form a complete orthonormal set in L2(G ):
f (x) =m−1∑k=0
〈f (·), e2πik·〉e2πikx =m−1∑k=0
f (k)e2πikx
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 6 / 36
![Page 35: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/35.jpg)
Orthogonality of characters on compact groups
If G is compact (=⇒ total Haar measure = 1) then characters are inL1(G ), being bounded.
If χ ∈ G then∫Gχ(x) dx =
∫Gχ(x + g) dx = χ(g)
∫Gχ(x) dx ,
so∫G χ = 0 if χ nontrivial, 1 if χ is trivial (= 1).
If χ, ψ ∈ G then χ(x)ψ(−x) is also a character. Hence
〈χ, ψ〉 =
∫Gχ(x)ψ(x) dx =
∫Gχ(x)ψ(−x) dx =
{1 χ = ψ0 χ 6= ψ
Fourier representation (inversion) in Zm: G = Zm =⇒ the m
characters form a complete orthonormal set in L2(G ):
f (x) =m−1∑k=0
〈f (·), e2πik·〉e2πikx =m−1∑k=0
f (k)e2πikx
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 6 / 36
![Page 36: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/36.jpg)
L2 of compact G
Trigonometric polynomials = finite linear combinations of characterson G
Example: G = T. Trig. polynomials are of the type∑N
k=−N cke2πikx .The least such N is called the degree of the polynomial.
Example: G = R. Trig. polynomials are of the type∑K
k=1 cke2πiλkx ,where λj ∈ R.
Compact G : Stone - Weierstrass Theorem =⇒ trig. polynomialsdense in C (G ) (in ‖·‖∞).
Fourier representation in L2(G ): Compact G : The characters form a
complete ONS. Since C (G ) is dense in L2(G ):
f =
∫χ∈bG f (χ)χ dχ all f ∈ L2(G ), convergence in L2(G )
G necessarily discrete in this case
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 7 / 36
![Page 37: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/37.jpg)
L2 of compact G
Trigonometric polynomials = finite linear combinations of characterson G
Example: G = T. Trig. polynomials are of the type∑N
k=−N cke2πikx .The least such N is called the degree of the polynomial.
Example: G = R. Trig. polynomials are of the type∑K
k=1 cke2πiλkx ,where λj ∈ R.
Compact G : Stone - Weierstrass Theorem =⇒ trig. polynomialsdense in C (G ) (in ‖·‖∞).
Fourier representation in L2(G ): Compact G : The characters form a
complete ONS. Since C (G ) is dense in L2(G ):
f =
∫χ∈bG f (χ)χ dχ all f ∈ L2(G ), convergence in L2(G )
G necessarily discrete in this case
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 7 / 36
![Page 38: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/38.jpg)
L2 of compact G
Trigonometric polynomials = finite linear combinations of characterson G
Example: G = T. Trig. polynomials are of the type∑N
k=−N cke2πikx .The least such N is called the degree of the polynomial.
Example: G = R. Trig. polynomials are of the type∑K
k=1 cke2πiλkx ,where λj ∈ R.
Compact G : Stone - Weierstrass Theorem =⇒ trig. polynomialsdense in C (G ) (in ‖·‖∞).
Fourier representation in L2(G ): Compact G : The characters form a
complete ONS. Since C (G ) is dense in L2(G ):
f =
∫χ∈bG f (χ)χ dχ all f ∈ L2(G ), convergence in L2(G )
G necessarily discrete in this case
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 7 / 36
![Page 39: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/39.jpg)
L2 of compact G
Trigonometric polynomials = finite linear combinations of characterson G
Example: G = T. Trig. polynomials are of the type∑N
k=−N cke2πikx .The least such N is called the degree of the polynomial.
Example: G = R. Trig. polynomials are of the type∑K
k=1 cke2πiλkx ,where λj ∈ R.
Compact G : Stone - Weierstrass Theorem =⇒ trig. polynomialsdense in C (G ) (in ‖·‖∞).
Fourier representation in L2(G ): Compact G : The characters form a
complete ONS. Since C (G ) is dense in L2(G ):
f =
∫χ∈bG f (χ)χ dχ all f ∈ L2(G ), convergence in L2(G )
G necessarily discrete in this case
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 7 / 36
![Page 40: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/40.jpg)
L2 of compact G
Trigonometric polynomials = finite linear combinations of characterson G
Example: G = T. Trig. polynomials are of the type∑N
k=−N cke2πikx .The least such N is called the degree of the polynomial.
Example: G = R. Trig. polynomials are of the type∑K
k=1 cke2πiλkx ,where λj ∈ R.
Compact G : Stone - Weierstrass Theorem =⇒ trig. polynomialsdense in C (G ) (in ‖·‖∞).
Fourier representation in L2(G ): Compact G : The characters form a
complete ONS. Since C (G ) is dense in L2(G ):
f =
∫χ∈bG f (χ)χ dχ all f ∈ L2(G ), convergence in L2(G )
G necessarily discrete in this case
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 7 / 36
![Page 41: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/41.jpg)
L2 of compact G
Trigonometric polynomials = finite linear combinations of characterson G
Example: G = T. Trig. polynomials are of the type∑N
k=−N cke2πikx .The least such N is called the degree of the polynomial.
Example: G = R. Trig. polynomials are of the type∑K
k=1 cke2πiλkx ,where λj ∈ R.
Compact G : Stone - Weierstrass Theorem =⇒ trig. polynomialsdense in C (G ) (in ‖·‖∞).
Fourier representation in L2(G ): Compact G : The characters form a
complete ONS. Since C (G ) is dense in L2(G ):
f =
∫χ∈bG f (χ)χ dχ all f ∈ L2(G ), convergence in L2(G )
G necessarily discrete in this case
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 7 / 36
![Page 42: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/42.jpg)
L2 of compact G , continued
Compact G : Parseval formula:∫G
f (x)g(x) dx =
∫bG f (χ)g(χ) dχ.
Compact G : f → f is an isometry from L2(G ) onto L2(G ).
Example: G = T∫T
f (x)g(x) dx =∑k∈Z
f (k)g(k), f , g ∈ L2(T).
Example: G = Zm
m−1∑j=0
f (j)g(j) =m−1∑k=0
f (k)g(k), all f , g : Zm → C
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 8 / 36
![Page 43: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/43.jpg)
L2 of compact G , continued
Compact G : Parseval formula:∫G
f (x)g(x) dx =
∫bG f (χ)g(χ) dχ.
Compact G : f → f is an isometry from L2(G ) onto L2(G ).
Example: G = T∫T
f (x)g(x) dx =∑k∈Z
f (k)g(k), f , g ∈ L2(T).
Example: G = Zm
m−1∑j=0
f (j)g(j) =m−1∑k=0
f (k)g(k), all f , g : Zm → C
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 8 / 36
![Page 44: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/44.jpg)
L2 of compact G , continued
Compact G : Parseval formula:∫G
f (x)g(x) dx =
∫bG f (χ)g(χ) dχ.
Compact G : f → f is an isometry from L2(G ) onto L2(G ).
Example: G = T∫T
f (x)g(x) dx =∑k∈Z
f (k)g(k), f , g ∈ L2(T).
Example: G = Zm
m−1∑j=0
f (j)g(j) =m−1∑k=0
f (k)g(k), all f , g : Zm → C
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 8 / 36
![Page 45: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/45.jpg)
L2 of compact G , continued
Compact G : Parseval formula:∫G
f (x)g(x) dx =
∫bG f (χ)g(χ) dχ.
Compact G : f → f is an isometry from L2(G ) onto L2(G ).
Example: G = T∫T
f (x)g(x) dx =∑k∈Z
f (k)g(k), f , g ∈ L2(T).
Example: G = Zm
m−1∑j=0
f (j)g(j) =m−1∑k=0
f (k)g(k), all f , g : Zm → C
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 8 / 36
![Page 46: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/46.jpg)
Triple correlations in Zp: an application
Problem of significance in (a) crystallography, (b) astrophysics:determine a subset E ⊆ Zn from its triple correlation:
NE (a, b) = #{x ∈ Zn : x , x + a, x + b ∈ E}, a, b ∈ Zn
=∑x∈Zn
1E (x)1E (x + a)1E (x + b)
Counts number of occurences of translated 3-point patterns {0, a, b}.
E can only be determined up to translation: E and E + t have thesame N(·, ·).For general n it has been proved that N(·, ·) cannot determine E evenup to translation (non-trivial).Special case: E can be determined up to translation from N(·, ·) ifn = p is a prime.Fourier transform of NE : Zn × Zn → R is easily computed:
NE (ξ, η) = 1E (ξ)1E (η)1E (−(ξ + η)), ξ, η ∈ Zn.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 9 / 36
![Page 47: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/47.jpg)
Triple correlations in Zp: an application
Problem of significance in (a) crystallography, (b) astrophysics:determine a subset E ⊆ Zn from its triple correlation:
NE (a, b) = #{x ∈ Zn : x , x + a, x + b ∈ E}, a, b ∈ Zn
=∑x∈Zn
1E (x)1E (x + a)1E (x + b)
Counts number of occurences of translated 3-point patterns {0, a, b}.E can only be determined up to translation: E and E + t have thesame N(·, ·).
For general n it has been proved that N(·, ·) cannot determine E evenup to translation (non-trivial).Special case: E can be determined up to translation from N(·, ·) ifn = p is a prime.Fourier transform of NE : Zn × Zn → R is easily computed:
NE (ξ, η) = 1E (ξ)1E (η)1E (−(ξ + η)), ξ, η ∈ Zn.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 9 / 36
![Page 48: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/48.jpg)
Triple correlations in Zp: an application
Problem of significance in (a) crystallography, (b) astrophysics:determine a subset E ⊆ Zn from its triple correlation:
NE (a, b) = #{x ∈ Zn : x , x + a, x + b ∈ E}, a, b ∈ Zn
=∑x∈Zn
1E (x)1E (x + a)1E (x + b)
Counts number of occurences of translated 3-point patterns {0, a, b}.E can only be determined up to translation: E and E + t have thesame N(·, ·).For general n it has been proved that N(·, ·) cannot determine E evenup to translation (non-trivial).
Special case: E can be determined up to translation from N(·, ·) ifn = p is a prime.Fourier transform of NE : Zn × Zn → R is easily computed:
NE (ξ, η) = 1E (ξ)1E (η)1E (−(ξ + η)), ξ, η ∈ Zn.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 9 / 36
![Page 49: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/49.jpg)
Triple correlations in Zp: an application
Problem of significance in (a) crystallography, (b) astrophysics:determine a subset E ⊆ Zn from its triple correlation:
NE (a, b) = #{x ∈ Zn : x , x + a, x + b ∈ E}, a, b ∈ Zn
=∑x∈Zn
1E (x)1E (x + a)1E (x + b)
Counts number of occurences of translated 3-point patterns {0, a, b}.E can only be determined up to translation: E and E + t have thesame N(·, ·).For general n it has been proved that N(·, ·) cannot determine E evenup to translation (non-trivial).Special case: E can be determined up to translation from N(·, ·) ifn = p is a prime.
Fourier transform of NE : Zn × Zn → R is easily computed:
NE (ξ, η) = 1E (ξ)1E (η)1E (−(ξ + η)), ξ, η ∈ Zn.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 9 / 36
![Page 50: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/50.jpg)
Triple correlations in Zp: an application
Problem of significance in (a) crystallography, (b) astrophysics:determine a subset E ⊆ Zn from its triple correlation:
NE (a, b) = #{x ∈ Zn : x , x + a, x + b ∈ E}, a, b ∈ Zn
=∑x∈Zn
1E (x)1E (x + a)1E (x + b)
Counts number of occurences of translated 3-point patterns {0, a, b}.E can only be determined up to translation: E and E + t have thesame N(·, ·).For general n it has been proved that N(·, ·) cannot determine E evenup to translation (non-trivial).Special case: E can be determined up to translation from N(·, ·) ifn = p is a prime.Fourier transform of NE : Zn × Zn → R is easily computed:
NE (ξ, η) = 1E (ξ)1E (η)1E (−(ξ + η)), ξ, η ∈ Zn.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 9 / 36
![Page 51: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/51.jpg)
Triple correlations in Zp: an application (continued)
If NE ≡ NF for E ,F ⊆ Zn then
1E (ξ)1E (η)1E (−(ξ+η)) = 1F (ξ)1F (η)1F (−(ξ+η)), ξ, η ∈ Zn (1)
Setting ξ = η = 0 we deduce #E = #F .
Setting η = 0, and using f (−x) = f (x) for real f , we get∣∣∣1E
∣∣∣ ≡ ∣∣∣1F
∣∣∣.If 1F is never 0 we divide (1) by its RHS to get
φ(ξ)φ(η) = φ(ξ + η), where φ = 1E
/1F (2)
Hence φ : Zn → C is a character and 1E ≡ φ1F .
Since Zn = Zn we have φ(ξ) = e2πitξ/n for some t ∈ Zn
Hence E = F + t
So NE determines E up to translation if 1E is never 0
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 10 / 36
![Page 52: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/52.jpg)
Triple correlations in Zp: an application (continued)
If NE ≡ NF for E ,F ⊆ Zn then
1E (ξ)1E (η)1E (−(ξ+η)) = 1F (ξ)1F (η)1F (−(ξ+η)), ξ, η ∈ Zn (1)
Setting ξ = η = 0 we deduce #E = #F .
Setting η = 0, and using f (−x) = f (x) for real f , we get∣∣∣1E
∣∣∣ ≡ ∣∣∣1F
∣∣∣.If 1F is never 0 we divide (1) by its RHS to get
φ(ξ)φ(η) = φ(ξ + η), where φ = 1E
/1F (2)
Hence φ : Zn → C is a character and 1E ≡ φ1F .
Since Zn = Zn we have φ(ξ) = e2πitξ/n for some t ∈ Zn
Hence E = F + t
So NE determines E up to translation if 1E is never 0
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 10 / 36
![Page 53: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/53.jpg)
Triple correlations in Zp: an application (continued)
If NE ≡ NF for E ,F ⊆ Zn then
1E (ξ)1E (η)1E (−(ξ+η)) = 1F (ξ)1F (η)1F (−(ξ+η)), ξ, η ∈ Zn (1)
Setting ξ = η = 0 we deduce #E = #F .
Setting η = 0, and using f (−x) = f (x) for real f , we get∣∣∣1E
∣∣∣ ≡ ∣∣∣1F
∣∣∣.
If 1F is never 0 we divide (1) by its RHS to get
φ(ξ)φ(η) = φ(ξ + η), where φ = 1E
/1F (2)
Hence φ : Zn → C is a character and 1E ≡ φ1F .
Since Zn = Zn we have φ(ξ) = e2πitξ/n for some t ∈ Zn
Hence E = F + t
So NE determines E up to translation if 1E is never 0
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 10 / 36
![Page 54: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/54.jpg)
Triple correlations in Zp: an application (continued)
If NE ≡ NF for E ,F ⊆ Zn then
1E (ξ)1E (η)1E (−(ξ+η)) = 1F (ξ)1F (η)1F (−(ξ+η)), ξ, η ∈ Zn (1)
Setting ξ = η = 0 we deduce #E = #F .
Setting η = 0, and using f (−x) = f (x) for real f , we get∣∣∣1E
∣∣∣ ≡ ∣∣∣1F
∣∣∣.If 1F is never 0 we divide (1) by its RHS to get
φ(ξ)φ(η) = φ(ξ + η), where φ = 1E
/1F (2)
Hence φ : Zn → C is a character and 1E ≡ φ1F .
Since Zn = Zn we have φ(ξ) = e2πitξ/n for some t ∈ Zn
Hence E = F + t
So NE determines E up to translation if 1E is never 0
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 10 / 36
![Page 55: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/55.jpg)
Triple correlations in Zp: an application (continued)
If NE ≡ NF for E ,F ⊆ Zn then
1E (ξ)1E (η)1E (−(ξ+η)) = 1F (ξ)1F (η)1F (−(ξ+η)), ξ, η ∈ Zn (1)
Setting ξ = η = 0 we deduce #E = #F .
Setting η = 0, and using f (−x) = f (x) for real f , we get∣∣∣1E
∣∣∣ ≡ ∣∣∣1F
∣∣∣.If 1F is never 0 we divide (1) by its RHS to get
φ(ξ)φ(η) = φ(ξ + η), where φ = 1E
/1F (2)
Hence φ : Zn → C is a character and 1E ≡ φ1F .
Since Zn = Zn we have φ(ξ) = e2πitξ/n for some t ∈ Zn
Hence E = F + t
So NE determines E up to translation if 1E is never 0
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 10 / 36
![Page 56: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/56.jpg)
Triple correlations in Zp: an application (continued)
If NE ≡ NF for E ,F ⊆ Zn then
1E (ξ)1E (η)1E (−(ξ+η)) = 1F (ξ)1F (η)1F (−(ξ+η)), ξ, η ∈ Zn (1)
Setting ξ = η = 0 we deduce #E = #F .
Setting η = 0, and using f (−x) = f (x) for real f , we get∣∣∣1E
∣∣∣ ≡ ∣∣∣1F
∣∣∣.If 1F is never 0 we divide (1) by its RHS to get
φ(ξ)φ(η) = φ(ξ + η), where φ = 1E
/1F (2)
Hence φ : Zn → C is a character and 1E ≡ φ1F .
Since Zn = Zn we have φ(ξ) = e2πitξ/n for some t ∈ Zn
Hence E = F + t
So NE determines E up to translation if 1E is never 0
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 10 / 36
![Page 57: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/57.jpg)
Triple correlations in Zp: an application (continued)
If NE ≡ NF for E ,F ⊆ Zn then
1E (ξ)1E (η)1E (−(ξ+η)) = 1F (ξ)1F (η)1F (−(ξ+η)), ξ, η ∈ Zn (1)
Setting ξ = η = 0 we deduce #E = #F .
Setting η = 0, and using f (−x) = f (x) for real f , we get∣∣∣1E
∣∣∣ ≡ ∣∣∣1F
∣∣∣.If 1F is never 0 we divide (1) by its RHS to get
φ(ξ)φ(η) = φ(ξ + η), where φ = 1E
/1F (2)
Hence φ : Zn → C is a character and 1E ≡ φ1F .
Since Zn = Zn we have φ(ξ) = e2πitξ/n for some t ∈ Zn
Hence E = F + t
So NE determines E up to translation if 1E is never 0
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 10 / 36
![Page 58: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/58.jpg)
Triple correlations in Zp: an application (continued)
If NE ≡ NF for E ,F ⊆ Zn then
1E (ξ)1E (η)1E (−(ξ+η)) = 1F (ξ)1F (η)1F (−(ξ+η)), ξ, η ∈ Zn (1)
Setting ξ = η = 0 we deduce #E = #F .
Setting η = 0, and using f (−x) = f (x) for real f , we get∣∣∣1E
∣∣∣ ≡ ∣∣∣1F
∣∣∣.If 1F is never 0 we divide (1) by its RHS to get
φ(ξ)φ(η) = φ(ξ + η), where φ = 1E
/1F (2)
Hence φ : Zn → C is a character and 1E ≡ φ1F .
Since Zn = Zn we have φ(ξ) = e2πitξ/n for some t ∈ Zn
Hence E = F + t
So NE determines E up to translation if 1E is never 0
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 10 / 36
![Page 59: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/59.jpg)
Triple correlations in Zp: an application (conclusion)
Suppose n = p is a prime, E ⊆ Zp. Then
1E (ξ) =1
p
∑s∈E
(ζξ)s , ζ = e−2πi/p is a p-root of unity. (3)
Each ζξ, ξ 6= 0, is a primitive p-th root of unity itself.
All powers (ζξ)s are distinct, so 1E (ξ) is a subset sum of all primitivep-th roots of unity (ξ 6= 0).
The polynomial 1 + x + x2 + · · ·+ xp−1 is the minimal polynomialover Q of each primitive root of unity (there are p − 1 of them).
It divides any polynomial in Q[x ] which vanishes on some primitivep-th root of unity
The only subset sums of all roots of unity which vanish are the emptyand the full sum (E = or E = Zp).
So in Zp the triple correlation NE (·, ·) determines E up to translation.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 11 / 36
![Page 60: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/60.jpg)
Triple correlations in Zp: an application (conclusion)
Suppose n = p is a prime, E ⊆ Zp. Then
1E (ξ) =1
p
∑s∈E
(ζξ)s , ζ = e−2πi/p is a p-root of unity. (3)
Each ζξ, ξ 6= 0, is a primitive p-th root of unity itself.
All powers (ζξ)s are distinct, so 1E (ξ) is a subset sum of all primitivep-th roots of unity (ξ 6= 0).
The polynomial 1 + x + x2 + · · ·+ xp−1 is the minimal polynomialover Q of each primitive root of unity (there are p − 1 of them).
It divides any polynomial in Q[x ] which vanishes on some primitivep-th root of unity
The only subset sums of all roots of unity which vanish are the emptyand the full sum (E = or E = Zp).
So in Zp the triple correlation NE (·, ·) determines E up to translation.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 11 / 36
![Page 61: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/61.jpg)
Triple correlations in Zp: an application (conclusion)
Suppose n = p is a prime, E ⊆ Zp. Then
1E (ξ) =1
p
∑s∈E
(ζξ)s , ζ = e−2πi/p is a p-root of unity. (3)
Each ζξ, ξ 6= 0, is a primitive p-th root of unity itself.
All powers (ζξ)s are distinct, so 1E (ξ) is a subset sum of all primitivep-th roots of unity (ξ 6= 0).
The polynomial 1 + x + x2 + · · ·+ xp−1 is the minimal polynomialover Q of each primitive root of unity (there are p − 1 of them).
It divides any polynomial in Q[x ] which vanishes on some primitivep-th root of unity
The only subset sums of all roots of unity which vanish are the emptyand the full sum (E = or E = Zp).
So in Zp the triple correlation NE (·, ·) determines E up to translation.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 11 / 36
![Page 62: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/62.jpg)
Triple correlations in Zp: an application (conclusion)
Suppose n = p is a prime, E ⊆ Zp. Then
1E (ξ) =1
p
∑s∈E
(ζξ)s , ζ = e−2πi/p is a p-root of unity. (3)
Each ζξ, ξ 6= 0, is a primitive p-th root of unity itself.
All powers (ζξ)s are distinct, so 1E (ξ) is a subset sum of all primitivep-th roots of unity (ξ 6= 0).
The polynomial 1 + x + x2 + · · ·+ xp−1 is the minimal polynomialover Q of each primitive root of unity (there are p − 1 of them).
It divides any polynomial in Q[x ] which vanishes on some primitivep-th root of unity
The only subset sums of all roots of unity which vanish are the emptyand the full sum (E = or E = Zp).
So in Zp the triple correlation NE (·, ·) determines E up to translation.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 11 / 36
![Page 63: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/63.jpg)
Triple correlations in Zp: an application (conclusion)
Suppose n = p is a prime, E ⊆ Zp. Then
1E (ξ) =1
p
∑s∈E
(ζξ)s , ζ = e−2πi/p is a p-root of unity. (3)
Each ζξ, ξ 6= 0, is a primitive p-th root of unity itself.
All powers (ζξ)s are distinct, so 1E (ξ) is a subset sum of all primitivep-th roots of unity (ξ 6= 0).
The polynomial 1 + x + x2 + · · ·+ xp−1 is the minimal polynomialover Q of each primitive root of unity (there are p − 1 of them).
It divides any polynomial in Q[x ] which vanishes on some primitivep-th root of unity
The only subset sums of all roots of unity which vanish are the emptyand the full sum (E = or E = Zp).
So in Zp the triple correlation NE (·, ·) determines E up to translation.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 11 / 36
![Page 64: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/64.jpg)
Triple correlations in Zp: an application (conclusion)
Suppose n = p is a prime, E ⊆ Zp. Then
1E (ξ) =1
p
∑s∈E
(ζξ)s , ζ = e−2πi/p is a p-root of unity. (3)
Each ζξ, ξ 6= 0, is a primitive p-th root of unity itself.
All powers (ζξ)s are distinct, so 1E (ξ) is a subset sum of all primitivep-th roots of unity (ξ 6= 0).
The polynomial 1 + x + x2 + · · ·+ xp−1 is the minimal polynomialover Q of each primitive root of unity (there are p − 1 of them).
It divides any polynomial in Q[x ] which vanishes on some primitivep-th root of unity
The only subset sums of all roots of unity which vanish are the emptyand the full sum (E = or E = Zp).
So in Zp the triple correlation NE (·, ·) determines E up to translation.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 11 / 36
![Page 65: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/65.jpg)
Triple correlations in Zp: an application (conclusion)
Suppose n = p is a prime, E ⊆ Zp. Then
1E (ξ) =1
p
∑s∈E
(ζξ)s , ζ = e−2πi/p is a p-root of unity. (3)
Each ζξ, ξ 6= 0, is a primitive p-th root of unity itself.
All powers (ζξ)s are distinct, so 1E (ξ) is a subset sum of all primitivep-th roots of unity (ξ 6= 0).
The polynomial 1 + x + x2 + · · ·+ xp−1 is the minimal polynomialover Q of each primitive root of unity (there are p − 1 of them).
It divides any polynomial in Q[x ] which vanishes on some primitivep-th root of unity
The only subset sums of all roots of unity which vanish are the emptyand the full sum (E = or E = Zp).
So in Zp the triple correlation NE (·, ·) determines E up to translation.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 11 / 36
![Page 66: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/66.jpg)
The basics of the FT on the torus (circle) T = R/2πZ
1 ≤ p ≤ q ⇐⇒ Lq(T) ⊆ Lp(T): nested Lp spaces. True on compactgroups.
f ∈ L1(T): we write f (x) ∼∑∞
k=−∞ f (k)e2πikx to denote theFourier series of f . No claim of convergence is made.
The Fourier coefficients of f (x) = e2πikx is the sequence f (n) = δk,n.
The Fourier series of a trig. poly. f (x) =∑N
k=−N ake2πikx is thesequence . . . , 0, 0, a−N , a−N+1, . . . , a0, . . . , aN , 0, 0, . . ..
Symmetric partial sums of the Fourier series of f :SN(f ; x) =
∑Nk=−N f (k)e2πikx
From f ∗ g = f · g we get easily SN(f ; x) = f (x) ∗ DN(x), where
DN(x) =N∑
k=−N
e2πikx =sin 2π(N + 1
2)x
sinπx(Dirichlet kernel of order N)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 12 / 36
![Page 67: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/67.jpg)
The basics of the FT on the torus (circle) T = R/2πZ
1 ≤ p ≤ q ⇐⇒ Lq(T) ⊆ Lp(T): nested Lp spaces. True on compactgroups.
f ∈ L1(T): we write f (x) ∼∑∞
k=−∞ f (k)e2πikx to denote theFourier series of f . No claim of convergence is made.
The Fourier coefficients of f (x) = e2πikx is the sequence f (n) = δk,n.
The Fourier series of a trig. poly. f (x) =∑N
k=−N ake2πikx is thesequence . . . , 0, 0, a−N , a−N+1, . . . , a0, . . . , aN , 0, 0, . . ..
Symmetric partial sums of the Fourier series of f :SN(f ; x) =
∑Nk=−N f (k)e2πikx
From f ∗ g = f · g we get easily SN(f ; x) = f (x) ∗ DN(x), where
DN(x) =N∑
k=−N
e2πikx =sin 2π(N + 1
2)x
sinπx(Dirichlet kernel of order N)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 12 / 36
![Page 68: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/68.jpg)
The basics of the FT on the torus (circle) T = R/2πZ
1 ≤ p ≤ q ⇐⇒ Lq(T) ⊆ Lp(T): nested Lp spaces. True on compactgroups.
f ∈ L1(T): we write f (x) ∼∑∞
k=−∞ f (k)e2πikx to denote theFourier series of f . No claim of convergence is made.
The Fourier coefficients of f (x) = e2πikx is the sequence f (n) = δk,n.
The Fourier series of a trig. poly. f (x) =∑N
k=−N ake2πikx is thesequence . . . , 0, 0, a−N , a−N+1, . . . , a0, . . . , aN , 0, 0, . . ..
Symmetric partial sums of the Fourier series of f :SN(f ; x) =
∑Nk=−N f (k)e2πikx
From f ∗ g = f · g we get easily SN(f ; x) = f (x) ∗ DN(x), where
DN(x) =N∑
k=−N
e2πikx =sin 2π(N + 1
2)x
sinπx(Dirichlet kernel of order N)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 12 / 36
![Page 69: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/69.jpg)
The basics of the FT on the torus (circle) T = R/2πZ
1 ≤ p ≤ q ⇐⇒ Lq(T) ⊆ Lp(T): nested Lp spaces. True on compactgroups.
f ∈ L1(T): we write f (x) ∼∑∞
k=−∞ f (k)e2πikx to denote theFourier series of f . No claim of convergence is made.
The Fourier coefficients of f (x) = e2πikx is the sequence f (n) = δk,n.
The Fourier series of a trig. poly. f (x) =∑N
k=−N ake2πikx is thesequence . . . , 0, 0, a−N , a−N+1, . . . , a0, . . . , aN , 0, 0, . . ..
Symmetric partial sums of the Fourier series of f :SN(f ; x) =
∑Nk=−N f (k)e2πikx
From f ∗ g = f · g we get easily SN(f ; x) = f (x) ∗ DN(x), where
DN(x) =N∑
k=−N
e2πikx =sin 2π(N + 1
2)x
sinπx(Dirichlet kernel of order N)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 12 / 36
![Page 70: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/70.jpg)
The basics of the FT on the torus (circle) T = R/2πZ
1 ≤ p ≤ q ⇐⇒ Lq(T) ⊆ Lp(T): nested Lp spaces. True on compactgroups.
f ∈ L1(T): we write f (x) ∼∑∞
k=−∞ f (k)e2πikx to denote theFourier series of f . No claim of convergence is made.
The Fourier coefficients of f (x) = e2πikx is the sequence f (n) = δk,n.
The Fourier series of a trig. poly. f (x) =∑N
k=−N ake2πikx is thesequence . . . , 0, 0, a−N , a−N+1, . . . , a0, . . . , aN , 0, 0, . . ..
Symmetric partial sums of the Fourier series of f :SN(f ; x) =
∑Nk=−N f (k)e2πikx
From f ∗ g = f · g we get easily SN(f ; x) = f (x) ∗ DN(x), where
DN(x) =N∑
k=−N
e2πikx =sin 2π(N + 1
2)x
sinπx(Dirichlet kernel of order N)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 12 / 36
![Page 71: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/71.jpg)
The basics of the FT on the torus (circle) T = R/2πZ
1 ≤ p ≤ q ⇐⇒ Lq(T) ⊆ Lp(T): nested Lp spaces. True on compactgroups.
f ∈ L1(T): we write f (x) ∼∑∞
k=−∞ f (k)e2πikx to denote theFourier series of f . No claim of convergence is made.
The Fourier coefficients of f (x) = e2πikx is the sequence f (n) = δk,n.
The Fourier series of a trig. poly. f (x) =∑N
k=−N ake2πikx is thesequence . . . , 0, 0, a−N , a−N+1, . . . , a0, . . . , aN , 0, 0, . . ..
Symmetric partial sums of the Fourier series of f :SN(f ; x) =
∑Nk=−N f (k)e2πikx
From f ∗ g = f · g we get easily SN(f ; x) = f (x) ∗ DN(x), where
DN(x) =N∑
k=−N
e2πikx =sin 2π(N + 1
2)x
sinπx(Dirichlet kernel of order N)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 12 / 36
![Page 72: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/72.jpg)
The Dirichlet kernel
-5
0
5
10
15
20
25
-0.4 -0.2 0 0.2 0.4
The Dirichlet kernel DN(x) for N = 10
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 13 / 36
![Page 73: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/73.jpg)
Pointwise convergence
Important: ‖DN‖1 ≥ C log N, as N →∞
TN : f → SN(f ; x) = DN ∗ f (x) is a (continuous) linear functionalC (T) → C. From the inequality ‖DN ∗ f ‖∞ ≤ ‖DN‖1‖f ‖∞‖TN‖ = ‖DN‖1 is unbounded
Banach-Steinhaus (uniform boundedness principle) =⇒Given x there are many continuous functions f such that TN(f ) isunbounded
Consequence: In general SN(f ; x) does not converge pointwise tof (x), even for continuous f
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 14 / 36
![Page 74: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/74.jpg)
Pointwise convergence
Important: ‖DN‖1 ≥ C log N, as N →∞TN : f → SN(f ; x) = DN ∗ f (x) is a (continuous) linear functionalC (T) → C. From the inequality ‖DN ∗ f ‖∞ ≤ ‖DN‖1‖f ‖∞
‖TN‖ = ‖DN‖1 is unbounded
Banach-Steinhaus (uniform boundedness principle) =⇒Given x there are many continuous functions f such that TN(f ) isunbounded
Consequence: In general SN(f ; x) does not converge pointwise tof (x), even for continuous f
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 14 / 36
![Page 75: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/75.jpg)
Pointwise convergence
Important: ‖DN‖1 ≥ C log N, as N →∞TN : f → SN(f ; x) = DN ∗ f (x) is a (continuous) linear functionalC (T) → C. From the inequality ‖DN ∗ f ‖∞ ≤ ‖DN‖1‖f ‖∞‖TN‖ = ‖DN‖1 is unbounded
Banach-Steinhaus (uniform boundedness principle) =⇒Given x there are many continuous functions f such that TN(f ) isunbounded
Consequence: In general SN(f ; x) does not converge pointwise tof (x), even for continuous f
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 14 / 36
![Page 76: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/76.jpg)
Pointwise convergence
Important: ‖DN‖1 ≥ C log N, as N →∞TN : f → SN(f ; x) = DN ∗ f (x) is a (continuous) linear functionalC (T) → C. From the inequality ‖DN ∗ f ‖∞ ≤ ‖DN‖1‖f ‖∞‖TN‖ = ‖DN‖1 is unbounded
Banach-Steinhaus (uniform boundedness principle) =⇒Given x there are many continuous functions f such that TN(f ) isunbounded
Consequence: In general SN(f ; x) does not converge pointwise tof (x), even for continuous f
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 14 / 36
![Page 77: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/77.jpg)
Pointwise convergence
Important: ‖DN‖1 ≥ C log N, as N →∞TN : f → SN(f ; x) = DN ∗ f (x) is a (continuous) linear functionalC (T) → C. From the inequality ‖DN ∗ f ‖∞ ≤ ‖DN‖1‖f ‖∞‖TN‖ = ‖DN‖1 is unbounded
Banach-Steinhaus (uniform boundedness principle) =⇒Given x there are many continuous functions f such that TN(f ) isunbounded
Consequence: In general SN(f ; x) does not converge pointwise tof (x), even for continuous f
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 14 / 36
![Page 78: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/78.jpg)
Summability
Look at the arithmetical means of SN(f ; x)
σN(f ; x) =1
N + 1
N∑n=0
Sn(f ; x) = KN ∗ f (x)
The Fejer kernel KN(x) is the mean of the Dirichlet kernels
KN(x) =N∑
n=−N
(1− |n|
N + 1
)e2πinx =
1
N + 1
(sinπ(N + 1)x
sinπx
)2
≥ 0.
KN(x) is an approximate identity:
(a)∫
T KN(x) dx = KN(0) = 1,(b) ‖KN‖1 is bounded (‖KN‖1 = 1, from nonnegativity and (a)),(c) for any ε > 0 we have
∫|x |>ε |KN(x)| dx → 0, as N →∞
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 15 / 36
![Page 79: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/79.jpg)
Summability
Look at the arithmetical means of SN(f ; x)
σN(f ; x) =1
N + 1
N∑n=0
Sn(f ; x) = KN ∗ f (x)
The Fejer kernel KN(x) is the mean of the Dirichlet kernels
KN(x) =N∑
n=−N
(1− |n|
N + 1
)e2πinx =
1
N + 1
(sinπ(N + 1)x
sinπx
)2
≥ 0.
KN(x) is an approximate identity:
(a)∫
T KN(x) dx = KN(0) = 1,(b) ‖KN‖1 is bounded (‖KN‖1 = 1, from nonnegativity and (a)),(c) for any ε > 0 we have
∫|x |>ε |KN(x)| dx → 0, as N →∞
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 15 / 36
![Page 80: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/80.jpg)
Summability
Look at the arithmetical means of SN(f ; x)
σN(f ; x) =1
N + 1
N∑n=0
Sn(f ; x) = KN ∗ f (x)
The Fejer kernel KN(x) is the mean of the Dirichlet kernels
KN(x) =N∑
n=−N
(1− |n|
N + 1
)e2πinx =
1
N + 1
(sinπ(N + 1)x
sinπx
)2
≥ 0.
KN(x) is an approximate identity:
(a)∫
T KN(x) dx = KN(0) = 1,(b) ‖KN‖1 is bounded (‖KN‖1 = 1, from nonnegativity and (a)),(c) for any ε > 0 we have
∫|x |>ε |KN(x)| dx → 0, as N →∞
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 15 / 36
![Page 81: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/81.jpg)
The Fejer kernel
0
2
4
6
8
10
12
-0.4 -0.2 0 0.2 0.4
The Fej’er kernel DN(x) for N = 10
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 16 / 36
![Page 82: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/82.jpg)
Summability (continued)
KN approximate identity =⇒ KN ∗ f (x) → f (x), in some Banachspaces. These can be:
C (T) normed with ‖·‖∞: If f ∈ C (T) then σN(f ; x) → f (x)uniformly in T.
Lp(T), 1 ≤ p <∞: If f ∈ Lp(T) then ‖σN(f ; x)− f (x)‖p → 0
Cn(T), all n-times C -differentiable functions, normed with‖f ‖Cn =
∑nk=0
∥∥f (k)∥∥∞
Summability implies uniqueness: the Fourier series of f ∈ L1(T)determines the function.
Another consequence: trig. polynomials are dense inLp(T),C (T),Cn(T)
Another important summability kernel: the Poisson kernel
P(r , x) =∑k∈Z
rke2πikx , 0 < r < 1: absolute convergence obvious
Significant for the theory of analytic functions.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 17 / 36
![Page 83: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/83.jpg)
Summability (continued)
KN approximate identity =⇒ KN ∗ f (x) → f (x), in some Banachspaces. These can be:
C (T) normed with ‖·‖∞: If f ∈ C (T) then σN(f ; x) → f (x)uniformly in T.
Lp(T), 1 ≤ p <∞: If f ∈ Lp(T) then ‖σN(f ; x)− f (x)‖p → 0
Cn(T), all n-times C -differentiable functions, normed with‖f ‖Cn =
∑nk=0
∥∥f (k)∥∥∞
Summability implies uniqueness: the Fourier series of f ∈ L1(T)determines the function.
Another consequence: trig. polynomials are dense inLp(T),C (T),Cn(T)
Another important summability kernel: the Poisson kernel
P(r , x) =∑k∈Z
rke2πikx , 0 < r < 1: absolute convergence obvious
Significant for the theory of analytic functions.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 17 / 36
![Page 84: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/84.jpg)
Summability (continued)
KN approximate identity =⇒ KN ∗ f (x) → f (x), in some Banachspaces. These can be:
C (T) normed with ‖·‖∞: If f ∈ C (T) then σN(f ; x) → f (x)uniformly in T.
Lp(T), 1 ≤ p <∞: If f ∈ Lp(T) then ‖σN(f ; x)− f (x)‖p → 0
Cn(T), all n-times C -differentiable functions, normed with‖f ‖Cn =
∑nk=0
∥∥f (k)∥∥∞
Summability implies uniqueness: the Fourier series of f ∈ L1(T)determines the function.
Another consequence: trig. polynomials are dense inLp(T),C (T),Cn(T)
Another important summability kernel: the Poisson kernel
P(r , x) =∑k∈Z
rke2πikx , 0 < r < 1: absolute convergence obvious
Significant for the theory of analytic functions.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 17 / 36
![Page 85: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/85.jpg)
Summability (continued)
KN approximate identity =⇒ KN ∗ f (x) → f (x), in some Banachspaces. These can be:
C (T) normed with ‖·‖∞: If f ∈ C (T) then σN(f ; x) → f (x)uniformly in T.
Lp(T), 1 ≤ p <∞: If f ∈ Lp(T) then ‖σN(f ; x)− f (x)‖p → 0
Cn(T), all n-times C -differentiable functions, normed with‖f ‖Cn =
∑nk=0
∥∥f (k)∥∥∞
Summability implies uniqueness: the Fourier series of f ∈ L1(T)determines the function.
Another consequence: trig. polynomials are dense inLp(T),C (T),Cn(T)
Another important summability kernel: the Poisson kernel
P(r , x) =∑k∈Z
rke2πikx , 0 < r < 1: absolute convergence obvious
Significant for the theory of analytic functions.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 17 / 36
![Page 86: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/86.jpg)
Summability (continued)
KN approximate identity =⇒ KN ∗ f (x) → f (x), in some Banachspaces. These can be:
C (T) normed with ‖·‖∞: If f ∈ C (T) then σN(f ; x) → f (x)uniformly in T.
Lp(T), 1 ≤ p <∞: If f ∈ Lp(T) then ‖σN(f ; x)− f (x)‖p → 0
Cn(T), all n-times C -differentiable functions, normed with‖f ‖Cn =
∑nk=0
∥∥f (k)∥∥∞
Summability implies uniqueness: the Fourier series of f ∈ L1(T)determines the function.
Another consequence: trig. polynomials are dense inLp(T),C (T),Cn(T)
Another important summability kernel: the Poisson kernel
P(r , x) =∑k∈Z
rke2πikx , 0 < r < 1: absolute convergence obvious
Significant for the theory of analytic functions.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 17 / 36
![Page 87: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/87.jpg)
Summability (continued)
KN approximate identity =⇒ KN ∗ f (x) → f (x), in some Banachspaces. These can be:
C (T) normed with ‖·‖∞: If f ∈ C (T) then σN(f ; x) → f (x)uniformly in T.
Lp(T), 1 ≤ p <∞: If f ∈ Lp(T) then ‖σN(f ; x)− f (x)‖p → 0
Cn(T), all n-times C -differentiable functions, normed with‖f ‖Cn =
∑nk=0
∥∥f (k)∥∥∞
Summability implies uniqueness: the Fourier series of f ∈ L1(T)determines the function.
Another consequence: trig. polynomials are dense inLp(T),C (T),Cn(T)
Another important summability kernel: the Poisson kernel
P(r , x) =∑k∈Z
rke2πikx , 0 < r < 1: absolute convergence obvious
Significant for the theory of analytic functions.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 17 / 36
![Page 88: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/88.jpg)
Summability (continued)
KN approximate identity =⇒ KN ∗ f (x) → f (x), in some Banachspaces. These can be:
C (T) normed with ‖·‖∞: If f ∈ C (T) then σN(f ; x) → f (x)uniformly in T.
Lp(T), 1 ≤ p <∞: If f ∈ Lp(T) then ‖σN(f ; x)− f (x)‖p → 0
Cn(T), all n-times C -differentiable functions, normed with‖f ‖Cn =
∑nk=0
∥∥f (k)∥∥∞
Summability implies uniqueness: the Fourier series of f ∈ L1(T)determines the function.
Another consequence: trig. polynomials are dense inLp(T),C (T),Cn(T)
Another important summability kernel: the Poisson kernel
P(r , x) =∑k∈Z
rke2πikx , 0 < r < 1: absolute convergence obvious
Significant for the theory of analytic functions.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 17 / 36
![Page 89: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/89.jpg)
The decay of the Fourier coefficients at ∞
Obvious: f (n) ≤ ‖f ‖1
Riemann-Lebesgue Lemma: lim|n|→∞ f (n) = 0 if f ∈ L1(T).Obviously true for trig. polynomials and they are dense in L1(T).
Can go to 0 arbitrarily slowly if we only assume f ∈ L1.
f (x) =∫ x0 g(t) dt, where
∫g = 0: f (n) = 1
2πin g(n) (Fubini)
Previous implies: f (|n|) = −f (−|n|) ≥ 0 =⇒∑
n 6=0 f (n)/n <∞.∑
n>0sin nxlog n is not a Fourier series.
f is an integral =⇒ f (n) = o(1/n): the “smoother” f is the betterdecay for the FT of f
f ∈ C 2(T) =⇒ absolute convergence for the Fourier Series of f .
Another condition that imposes “decay”:
f ∈ L2(T) =⇒∑
n
∣∣∣f (n)∣∣∣2 <∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 18 / 36
![Page 90: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/90.jpg)
The decay of the Fourier coefficients at ∞
Obvious: f (n) ≤ ‖f ‖1
Riemann-Lebesgue Lemma: lim|n|→∞ f (n) = 0 if f ∈ L1(T).Obviously true for trig. polynomials and they are dense in L1(T).
Can go to 0 arbitrarily slowly if we only assume f ∈ L1.
f (x) =∫ x0 g(t) dt, where
∫g = 0: f (n) = 1
2πin g(n) (Fubini)
Previous implies: f (|n|) = −f (−|n|) ≥ 0 =⇒∑
n 6=0 f (n)/n <∞.∑
n>0sin nxlog n is not a Fourier series.
f is an integral =⇒ f (n) = o(1/n): the “smoother” f is the betterdecay for the FT of f
f ∈ C 2(T) =⇒ absolute convergence for the Fourier Series of f .
Another condition that imposes “decay”:
f ∈ L2(T) =⇒∑
n
∣∣∣f (n)∣∣∣2 <∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 18 / 36
![Page 91: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/91.jpg)
The decay of the Fourier coefficients at ∞
Obvious: f (n) ≤ ‖f ‖1
Riemann-Lebesgue Lemma: lim|n|→∞ f (n) = 0 if f ∈ L1(T).Obviously true for trig. polynomials and they are dense in L1(T).
Can go to 0 arbitrarily slowly if we only assume f ∈ L1.
f (x) =∫ x0 g(t) dt, where
∫g = 0: f (n) = 1
2πin g(n) (Fubini)
Previous implies: f (|n|) = −f (−|n|) ≥ 0 =⇒∑
n 6=0 f (n)/n <∞.∑
n>0sin nxlog n is not a Fourier series.
f is an integral =⇒ f (n) = o(1/n): the “smoother” f is the betterdecay for the FT of f
f ∈ C 2(T) =⇒ absolute convergence for the Fourier Series of f .
Another condition that imposes “decay”:
f ∈ L2(T) =⇒∑
n
∣∣∣f (n)∣∣∣2 <∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 18 / 36
![Page 92: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/92.jpg)
The decay of the Fourier coefficients at ∞
Obvious: f (n) ≤ ‖f ‖1
Riemann-Lebesgue Lemma: lim|n|→∞ f (n) = 0 if f ∈ L1(T).Obviously true for trig. polynomials and they are dense in L1(T).
Can go to 0 arbitrarily slowly if we only assume f ∈ L1.
f (x) =∫ x0 g(t) dt, where
∫g = 0: f (n) = 1
2πin g(n) (Fubini)
Previous implies: f (|n|) = −f (−|n|) ≥ 0 =⇒∑
n 6=0 f (n)/n <∞.∑
n>0sin nxlog n is not a Fourier series.
f is an integral =⇒ f (n) = o(1/n): the “smoother” f is the betterdecay for the FT of f
f ∈ C 2(T) =⇒ absolute convergence for the Fourier Series of f .
Another condition that imposes “decay”:
f ∈ L2(T) =⇒∑
n
∣∣∣f (n)∣∣∣2 <∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 18 / 36
![Page 93: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/93.jpg)
The decay of the Fourier coefficients at ∞
Obvious: f (n) ≤ ‖f ‖1
Riemann-Lebesgue Lemma: lim|n|→∞ f (n) = 0 if f ∈ L1(T).Obviously true for trig. polynomials and they are dense in L1(T).
Can go to 0 arbitrarily slowly if we only assume f ∈ L1.
f (x) =∫ x0 g(t) dt, where
∫g = 0: f (n) = 1
2πin g(n) (Fubini)
Previous implies: f (|n|) = −f (−|n|) ≥ 0 =⇒∑
n 6=0 f (n)/n <∞.
∑n>0
sin nxlog n is not a Fourier series.
f is an integral =⇒ f (n) = o(1/n): the “smoother” f is the betterdecay for the FT of f
f ∈ C 2(T) =⇒ absolute convergence for the Fourier Series of f .
Another condition that imposes “decay”:
f ∈ L2(T) =⇒∑
n
∣∣∣f (n)∣∣∣2 <∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 18 / 36
![Page 94: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/94.jpg)
The decay of the Fourier coefficients at ∞
Obvious: f (n) ≤ ‖f ‖1
Riemann-Lebesgue Lemma: lim|n|→∞ f (n) = 0 if f ∈ L1(T).Obviously true for trig. polynomials and they are dense in L1(T).
Can go to 0 arbitrarily slowly if we only assume f ∈ L1.
f (x) =∫ x0 g(t) dt, where
∫g = 0: f (n) = 1
2πin g(n) (Fubini)
Previous implies: f (|n|) = −f (−|n|) ≥ 0 =⇒∑
n 6=0 f (n)/n <∞.∑
n>0sin nxlog n is not a Fourier series.
f is an integral =⇒ f (n) = o(1/n): the “smoother” f is the betterdecay for the FT of f
f ∈ C 2(T) =⇒ absolute convergence for the Fourier Series of f .
Another condition that imposes “decay”:
f ∈ L2(T) =⇒∑
n
∣∣∣f (n)∣∣∣2 <∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 18 / 36
![Page 95: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/95.jpg)
The decay of the Fourier coefficients at ∞
Obvious: f (n) ≤ ‖f ‖1
Riemann-Lebesgue Lemma: lim|n|→∞ f (n) = 0 if f ∈ L1(T).Obviously true for trig. polynomials and they are dense in L1(T).
Can go to 0 arbitrarily slowly if we only assume f ∈ L1.
f (x) =∫ x0 g(t) dt, where
∫g = 0: f (n) = 1
2πin g(n) (Fubini)
Previous implies: f (|n|) = −f (−|n|) ≥ 0 =⇒∑
n 6=0 f (n)/n <∞.∑
n>0sin nxlog n is not a Fourier series.
f is an integral =⇒ f (n) = o(1/n): the “smoother” f is the betterdecay for the FT of f
f ∈ C 2(T) =⇒ absolute convergence for the Fourier Series of f .
Another condition that imposes “decay”:
f ∈ L2(T) =⇒∑
n
∣∣∣f (n)∣∣∣2 <∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 18 / 36
![Page 96: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/96.jpg)
The decay of the Fourier coefficients at ∞
Obvious: f (n) ≤ ‖f ‖1
Riemann-Lebesgue Lemma: lim|n|→∞ f (n) = 0 if f ∈ L1(T).Obviously true for trig. polynomials and they are dense in L1(T).
Can go to 0 arbitrarily slowly if we only assume f ∈ L1.
f (x) =∫ x0 g(t) dt, where
∫g = 0: f (n) = 1
2πin g(n) (Fubini)
Previous implies: f (|n|) = −f (−|n|) ≥ 0 =⇒∑
n 6=0 f (n)/n <∞.∑
n>0sin nxlog n is not a Fourier series.
f is an integral =⇒ f (n) = o(1/n): the “smoother” f is the betterdecay for the FT of f
f ∈ C 2(T) =⇒ absolute convergence for the Fourier Series of f .
Another condition that imposes “decay”:
f ∈ L2(T) =⇒∑
n
∣∣∣f (n)∣∣∣2 <∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 18 / 36
![Page 97: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/97.jpg)
The decay of the Fourier coefficients at ∞
Obvious: f (n) ≤ ‖f ‖1
Riemann-Lebesgue Lemma: lim|n|→∞ f (n) = 0 if f ∈ L1(T).Obviously true for trig. polynomials and they are dense in L1(T).
Can go to 0 arbitrarily slowly if we only assume f ∈ L1.
f (x) =∫ x0 g(t) dt, where
∫g = 0: f (n) = 1
2πin g(n) (Fubini)
Previous implies: f (|n|) = −f (−|n|) ≥ 0 =⇒∑
n 6=0 f (n)/n <∞.∑
n>0sin nxlog n is not a Fourier series.
f is an integral =⇒ f (n) = o(1/n): the “smoother” f is the betterdecay for the FT of f
f ∈ C 2(T) =⇒ absolute convergence for the Fourier Series of f .
Another condition that imposes “decay”:
f ∈ L2(T) =⇒∑
n
∣∣∣f (n)∣∣∣2 <∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 18 / 36
![Page 98: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/98.jpg)
Interpolation of operators
T is bounded linear operator on dense subsets of Lp1 and Lp2 :
‖Tf ‖q1≤ C1‖f ‖p1
, ‖Tf ‖q2≤ C2‖f ‖p2
Riesz-Thorin interpolation theorem: T : Lp → Lq for any pbetween p1, p2 (all p’s and q’s ≥ 1).
p and q are related by:
1
p= t
1
p1+ (1− t)
1
p2=⇒ 1
q= t
1
q1+ (1− t)
1
q2
‖T‖Lp→Lq ≤ C t1C
(1−t)2
The exponents p, q, . . . are allowed to be ∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 19 / 36
![Page 99: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/99.jpg)
Interpolation of operators
T is bounded linear operator on dense subsets of Lp1 and Lp2 :
‖Tf ‖q1≤ C1‖f ‖p1
, ‖Tf ‖q2≤ C2‖f ‖p2
Riesz-Thorin interpolation theorem: T : Lp → Lq for any pbetween p1, p2 (all p’s and q’s ≥ 1).
p and q are related by:
1
p= t
1
p1+ (1− t)
1
p2=⇒ 1
q= t
1
q1+ (1− t)
1
q2
‖T‖Lp→Lq ≤ C t1C
(1−t)2
The exponents p, q, . . . are allowed to be ∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 19 / 36
![Page 100: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/100.jpg)
Interpolation of operators
T is bounded linear operator on dense subsets of Lp1 and Lp2 :
‖Tf ‖q1≤ C1‖f ‖p1
, ‖Tf ‖q2≤ C2‖f ‖p2
Riesz-Thorin interpolation theorem: T : Lp → Lq for any pbetween p1, p2 (all p’s and q’s ≥ 1).
p and q are related by:
1
p= t
1
p1+ (1− t)
1
p2=⇒ 1
q= t
1
q1+ (1− t)
1
q2
‖T‖Lp→Lq ≤ C t1C
(1−t)2
The exponents p, q, . . . are allowed to be ∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 19 / 36
![Page 101: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/101.jpg)
Interpolation of operators
T is bounded linear operator on dense subsets of Lp1 and Lp2 :
‖Tf ‖q1≤ C1‖f ‖p1
, ‖Tf ‖q2≤ C2‖f ‖p2
Riesz-Thorin interpolation theorem: T : Lp → Lq for any pbetween p1, p2 (all p’s and q’s ≥ 1).
p and q are related by:
1
p= t
1
p1+ (1− t)
1
p2=⇒ 1
q= t
1
q1+ (1− t)
1
q2
‖T‖Lp→Lq ≤ C t1C
(1−t)2
The exponents p, q, . . . are allowed to be ∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 19 / 36
![Page 102: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/102.jpg)
Interpolation of operators
T is bounded linear operator on dense subsets of Lp1 and Lp2 :
‖Tf ‖q1≤ C1‖f ‖p1
, ‖Tf ‖q2≤ C2‖f ‖p2
Riesz-Thorin interpolation theorem: T : Lp → Lq for any pbetween p1, p2 (all p’s and q’s ≥ 1).
p and q are related by:
1
p= t
1
p1+ (1− t)
1
p2=⇒ 1
q= t
1
q1+ (1− t)
1
q2
‖T‖Lp→Lq ≤ C t1C
(1−t)2
The exponents p, q, . . . are allowed to be ∞.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 19 / 36
![Page 103: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/103.jpg)
Interpolation of operators: the 1/p, 1
/q plane
1/p
s
sS
SS
SS
SS
SS
SS
0 1
0
1
1/q
(1/p1, 1/q1)
(1/p2, 1/q2)
(1/p, 1/q)
s
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 20 / 36
![Page 104: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/104.jpg)
The Hausdorff-Young inequality
Hausdorff-Young: Suppose 1 ≤ p ≤ 2, 1p + 1
q = 1, andf ∈ Lp(T). It follows that∥∥∥f
∥∥∥Lq(Z)
≤ Cp‖f ‖Lp(T)
False if p > 2.
Clearly true if p = 1 (trivial) or p = 2 (Parseval).
Use Riesz-Thorin interpolation for 1 < p < 2 for the operatorf → f from Lp(T) → Lq(Z).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 21 / 36
![Page 105: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/105.jpg)
The Hausdorff-Young inequality
Hausdorff-Young: Suppose 1 ≤ p ≤ 2, 1p + 1
q = 1, andf ∈ Lp(T). It follows that∥∥∥f
∥∥∥Lq(Z)
≤ Cp‖f ‖Lp(T)
False if p > 2.
Clearly true if p = 1 (trivial) or p = 2 (Parseval).
Use Riesz-Thorin interpolation for 1 < p < 2 for the operatorf → f from Lp(T) → Lq(Z).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 21 / 36
![Page 106: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/106.jpg)
The Hausdorff-Young inequality
Hausdorff-Young: Suppose 1 ≤ p ≤ 2, 1p + 1
q = 1, andf ∈ Lp(T). It follows that∥∥∥f
∥∥∥Lq(Z)
≤ Cp‖f ‖Lp(T)
False if p > 2.
Clearly true if p = 1 (trivial) or p = 2 (Parseval).
Use Riesz-Thorin interpolation for 1 < p < 2 for the operatorf → f from Lp(T) → Lq(Z).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 21 / 36
![Page 107: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/107.jpg)
The Hausdorff-Young inequality
Hausdorff-Young: Suppose 1 ≤ p ≤ 2, 1p + 1
q = 1, andf ∈ Lp(T). It follows that∥∥∥f
∥∥∥Lq(Z)
≤ Cp‖f ‖Lp(T)
False if p > 2.
Clearly true if p = 1 (trivial) or p = 2 (Parseval).
Use Riesz-Thorin interpolation for 1 < p < 2 for the operatorf → f from Lp(T) → Lq(Z).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 21 / 36
![Page 108: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/108.jpg)
An application: the isoperimetric inequality
Suppose Γ is a simple closed curve in the plane with perimeter Lenclosing area A.
A ≤ 1
4πL2 (isoperimetric inequality)
Equality holds only when Γ is a circle.
Wirtinger’s inequality: if f ∈ C∞(T) then∫ 1
0
∣∣∣f (x)− f (0)∣∣∣2 dx ≤ 1
4π2
∫ 1
0
∣∣f ′(x)∣∣2 dx . (4)
By smoothness f (x) equals its Fourier series and so doesf ′(x) = 2πi
∑n nf (n)e2πinx
FT is an isometry (Parseval) so LHS of (4) is∑
n 6=0
∣∣∣f (n)∣∣∣2 while the
RHS is∑
n 6=0 n∣∣∣f (n)
∣∣∣2 so (4) holds.
Equality in (4) precisely when f (x) = f (−1)e−2πix + f (0) + f (1)e2πix .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 22 / 36
![Page 109: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/109.jpg)
An application: the isoperimetric inequality
Suppose Γ is a simple closed curve in the plane with perimeter Lenclosing area A.
A ≤ 1
4πL2 (isoperimetric inequality)
Equality holds only when Γ is a circle.
Wirtinger’s inequality: if f ∈ C∞(T) then∫ 1
0
∣∣∣f (x)− f (0)∣∣∣2 dx ≤ 1
4π2
∫ 1
0
∣∣f ′(x)∣∣2 dx . (4)
By smoothness f (x) equals its Fourier series and so doesf ′(x) = 2πi
∑n nf (n)e2πinx
FT is an isometry (Parseval) so LHS of (4) is∑
n 6=0
∣∣∣f (n)∣∣∣2 while the
RHS is∑
n 6=0 n∣∣∣f (n)
∣∣∣2 so (4) holds.
Equality in (4) precisely when f (x) = f (−1)e−2πix + f (0) + f (1)e2πix .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 22 / 36
![Page 110: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/110.jpg)
An application: the isoperimetric inequality
Suppose Γ is a simple closed curve in the plane with perimeter Lenclosing area A.
A ≤ 1
4πL2 (isoperimetric inequality)
Equality holds only when Γ is a circle.
Wirtinger’s inequality: if f ∈ C∞(T) then∫ 1
0
∣∣∣f (x)− f (0)∣∣∣2 dx ≤ 1
4π2
∫ 1
0
∣∣f ′(x)∣∣2 dx . (4)
By smoothness f (x) equals its Fourier series and so doesf ′(x) = 2πi
∑n nf (n)e2πinx
FT is an isometry (Parseval) so LHS of (4) is∑
n 6=0
∣∣∣f (n)∣∣∣2 while the
RHS is∑
n 6=0 n∣∣∣f (n)
∣∣∣2 so (4) holds.
Equality in (4) precisely when f (x) = f (−1)e−2πix + f (0) + f (1)e2πix .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 22 / 36
![Page 111: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/111.jpg)
An application: the isoperimetric inequality
Suppose Γ is a simple closed curve in the plane with perimeter Lenclosing area A.
A ≤ 1
4πL2 (isoperimetric inequality)
Equality holds only when Γ is a circle.
Wirtinger’s inequality: if f ∈ C∞(T) then∫ 1
0
∣∣∣f (x)− f (0)∣∣∣2 dx ≤ 1
4π2
∫ 1
0
∣∣f ′(x)∣∣2 dx . (4)
By smoothness f (x) equals its Fourier series and so doesf ′(x) = 2πi
∑n nf (n)e2πinx
FT is an isometry (Parseval) so LHS of (4) is∑
n 6=0
∣∣∣f (n)∣∣∣2 while the
RHS is∑
n 6=0 n∣∣∣f (n)
∣∣∣2 so (4) holds.
Equality in (4) precisely when f (x) = f (−1)e−2πix + f (0) + f (1)e2πix .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 22 / 36
![Page 112: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/112.jpg)
An application: the isoperimetric inequality
Suppose Γ is a simple closed curve in the plane with perimeter Lenclosing area A.
A ≤ 1
4πL2 (isoperimetric inequality)
Equality holds only when Γ is a circle.
Wirtinger’s inequality: if f ∈ C∞(T) then∫ 1
0
∣∣∣f (x)− f (0)∣∣∣2 dx ≤ 1
4π2
∫ 1
0
∣∣f ′(x)∣∣2 dx . (4)
By smoothness f (x) equals its Fourier series and so doesf ′(x) = 2πi
∑n nf (n)e2πinx
FT is an isometry (Parseval) so LHS of (4) is∑
n 6=0
∣∣∣f (n)∣∣∣2 while the
RHS is∑
n 6=0 n∣∣∣f (n)
∣∣∣2 so (4) holds.
Equality in (4) precisely when f (x) = f (−1)e−2πix + f (0) + f (1)e2πix .
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 22 / 36
![Page 113: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/113.jpg)
An application: the isoperimetric inequality (continued)
Hurwitz’ proof. First assume Γ is smooth, has L = 1.
Parametrization of Γ: (x(s), y(s)), 0 ≤ s ≤ 1 w.r.t. arc length sx , y ∈ C∞(T), (x ′(s))2 + (y ′(s))2 = 1.
Green’s Theorem =⇒ area A =∫ 10 x(s)y ′(s) ds:
A =
∫(x(s)− x(0))y ′(s) =
=1
4π
∫(2π(x(s)− x(0)))2 + y ′(s)2 − (2π(x(s)− x(0))− y ′(s))2
≤ 1/4π
∫4π2(x(s)− x(0))2 + y ′(s)2 (drop last term)
≤ 1/4π
∫x ′(s)2 + y ′(s)2 (Wirtinger’s ineq)
= 1/4π
For equality must have x(s) = a cos 2πs + b sin 2πs + c ,y ′(s) = 2π(x(s)− x(0)). So x(s)2 + y(s)2 constant if c = 0.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 23 / 36
![Page 114: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/114.jpg)
An application: the isoperimetric inequality (continued)
Hurwitz’ proof. First assume Γ is smooth, has L = 1.Parametrization of Γ: (x(s), y(s)), 0 ≤ s ≤ 1 w.r.t. arc length s
x , y ∈ C∞(T), (x ′(s))2 + (y ′(s))2 = 1.
Green’s Theorem =⇒ area A =∫ 10 x(s)y ′(s) ds:
A =
∫(x(s)− x(0))y ′(s) =
=1
4π
∫(2π(x(s)− x(0)))2 + y ′(s)2 − (2π(x(s)− x(0))− y ′(s))2
≤ 1/4π
∫4π2(x(s)− x(0))2 + y ′(s)2 (drop last term)
≤ 1/4π
∫x ′(s)2 + y ′(s)2 (Wirtinger’s ineq)
= 1/4π
For equality must have x(s) = a cos 2πs + b sin 2πs + c ,y ′(s) = 2π(x(s)− x(0)). So x(s)2 + y(s)2 constant if c = 0.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 23 / 36
![Page 115: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/115.jpg)
An application: the isoperimetric inequality (continued)
Hurwitz’ proof. First assume Γ is smooth, has L = 1.Parametrization of Γ: (x(s), y(s)), 0 ≤ s ≤ 1 w.r.t. arc length sx , y ∈ C∞(T), (x ′(s))2 + (y ′(s))2 = 1.
Green’s Theorem =⇒ area A =∫ 10 x(s)y ′(s) ds:
A =
∫(x(s)− x(0))y ′(s) =
=1
4π
∫(2π(x(s)− x(0)))2 + y ′(s)2 − (2π(x(s)− x(0))− y ′(s))2
≤ 1/4π
∫4π2(x(s)− x(0))2 + y ′(s)2 (drop last term)
≤ 1/4π
∫x ′(s)2 + y ′(s)2 (Wirtinger’s ineq)
= 1/4π
For equality must have x(s) = a cos 2πs + b sin 2πs + c ,y ′(s) = 2π(x(s)− x(0)). So x(s)2 + y(s)2 constant if c = 0.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 23 / 36
![Page 116: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/116.jpg)
An application: the isoperimetric inequality (continued)
Hurwitz’ proof. First assume Γ is smooth, has L = 1.Parametrization of Γ: (x(s), y(s)), 0 ≤ s ≤ 1 w.r.t. arc length sx , y ∈ C∞(T), (x ′(s))2 + (y ′(s))2 = 1.
Green’s Theorem =⇒ area A =∫ 10 x(s)y ′(s) ds:
A =
∫(x(s)− x(0))y ′(s) =
=1
4π
∫(2π(x(s)− x(0)))2 + y ′(s)2 − (2π(x(s)− x(0))− y ′(s))2
≤ 1/4π
∫4π2(x(s)− x(0))2 + y ′(s)2 (drop last term)
≤ 1/4π
∫x ′(s)2 + y ′(s)2 (Wirtinger’s ineq)
= 1/4π
For equality must have x(s) = a cos 2πs + b sin 2πs + c ,y ′(s) = 2π(x(s)− x(0)). So x(s)2 + y(s)2 constant if c = 0.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 23 / 36
![Page 117: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/117.jpg)
An application: the isoperimetric inequality (continued)
Hurwitz’ proof. First assume Γ is smooth, has L = 1.Parametrization of Γ: (x(s), y(s)), 0 ≤ s ≤ 1 w.r.t. arc length sx , y ∈ C∞(T), (x ′(s))2 + (y ′(s))2 = 1.
Green’s Theorem =⇒ area A =∫ 10 x(s)y ′(s) ds:
A =
∫(x(s)− x(0))y ′(s) =
=1
4π
∫(2π(x(s)− x(0)))2 + y ′(s)2 − (2π(x(s)− x(0))− y ′(s))2
≤ 1/4π
∫4π2(x(s)− x(0))2 + y ′(s)2 (drop last term)
≤ 1/4π
∫x ′(s)2 + y ′(s)2 (Wirtinger’s ineq)
= 1/4π
For equality must have x(s) = a cos 2πs + b sin 2πs + c ,y ′(s) = 2π(x(s)− x(0)). So x(s)2 + y(s)2 constant if c = 0.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 23 / 36
![Page 118: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/118.jpg)
Fourier transform on Rn
Initially defined only for f ∈ L1(Rn). f (ξ) =∫
Rn f (x)e−2πiξ·x dx .
Follows:∥∥∥f
∥∥∥∞≤ ‖f ‖1. f is continuous.
Trig. polynomials are not dense anymore in the usual spaces.
But Riemann-Lebesgue is true. First for indicator function of aninterval
[a1, b1]× · · · × [an, bn].
Then approximate an L1 function by finite linear combinations ofsuch.
Multi-index notation α = (α1, . . . , αn) ∈ Nn:(a) |α| = α1 + · · ·+ αn.(b) xα = xα1
1 xα22 · · · xαn
n
(c) ∂α = (∂/∂1)
α1 · · · (∂/∂n)
αn
Diff operators D jφ := 12πi (∂
/∂xj), Dαφ = (1
/2πi)|α|∂α.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 24 / 36
![Page 119: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/119.jpg)
Fourier transform on Rn
Initially defined only for f ∈ L1(Rn). f (ξ) =∫
Rn f (x)e−2πiξ·x dx .
Follows:∥∥∥f
∥∥∥∞≤ ‖f ‖1. f is continuous.
Trig. polynomials are not dense anymore in the usual spaces.
But Riemann-Lebesgue is true. First for indicator function of aninterval
[a1, b1]× · · · × [an, bn].
Then approximate an L1 function by finite linear combinations ofsuch.
Multi-index notation α = (α1, . . . , αn) ∈ Nn:(a) |α| = α1 + · · ·+ αn.(b) xα = xα1
1 xα22 · · · xαn
n
(c) ∂α = (∂/∂1)
α1 · · · (∂/∂n)
αn
Diff operators D jφ := 12πi (∂
/∂xj), Dαφ = (1
/2πi)|α|∂α.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 24 / 36
![Page 120: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/120.jpg)
Fourier transform on Rn
Initially defined only for f ∈ L1(Rn). f (ξ) =∫
Rn f (x)e−2πiξ·x dx .
Follows:∥∥∥f
∥∥∥∞≤ ‖f ‖1. f is continuous.
Trig. polynomials are not dense anymore in the usual spaces.
But Riemann-Lebesgue is true. First for indicator function of aninterval
[a1, b1]× · · · × [an, bn].
Then approximate an L1 function by finite linear combinations ofsuch.
Multi-index notation α = (α1, . . . , αn) ∈ Nn:(a) |α| = α1 + · · ·+ αn.(b) xα = xα1
1 xα22 · · · xαn
n
(c) ∂α = (∂/∂1)
α1 · · · (∂/∂n)
αn
Diff operators D jφ := 12πi (∂
/∂xj), Dαφ = (1
/2πi)|α|∂α.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 24 / 36
![Page 121: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/121.jpg)
Fourier transform on Rn
Initially defined only for f ∈ L1(Rn). f (ξ) =∫
Rn f (x)e−2πiξ·x dx .
Follows:∥∥∥f
∥∥∥∞≤ ‖f ‖1. f is continuous.
Trig. polynomials are not dense anymore in the usual spaces.
But Riemann-Lebesgue is true. First for indicator function of aninterval
[a1, b1]× · · · × [an, bn].
Then approximate an L1 function by finite linear combinations ofsuch.
Multi-index notation α = (α1, . . . , αn) ∈ Nn:(a) |α| = α1 + · · ·+ αn.(b) xα = xα1
1 xα22 · · · xαn
n
(c) ∂α = (∂/∂1)
α1 · · · (∂/∂n)
αn
Diff operators D jφ := 12πi (∂
/∂xj), Dαφ = (1
/2πi)|α|∂α.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 24 / 36
![Page 122: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/122.jpg)
Fourier transform on Rn
Initially defined only for f ∈ L1(Rn). f (ξ) =∫
Rn f (x)e−2πiξ·x dx .
Follows:∥∥∥f
∥∥∥∞≤ ‖f ‖1. f is continuous.
Trig. polynomials are not dense anymore in the usual spaces.
But Riemann-Lebesgue is true. First for indicator function of aninterval
[a1, b1]× · · · × [an, bn].
Then approximate an L1 function by finite linear combinations ofsuch.
Multi-index notation α = (α1, . . . , αn) ∈ Nn:(a) |α| = α1 + · · ·+ αn.(b) xα = xα1
1 xα22 · · · xαn
n
(c) ∂α = (∂/∂1)
α1 · · · (∂/∂n)
αn
Diff operators D jφ := 12πi (∂
/∂xj), Dαφ = (1
/2πi)|α|∂α.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 24 / 36
![Page 123: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/123.jpg)
Schwartz functions on Rn
Lp(Rn) spaces are not nested.
Not clear how to define f for f ∈ L2.
Schwartz class S: those φ ∈ C∞(Rn) s.t. for all multiindices α, γ
‖φ‖α,γ := supx∈Rn
|xγ∂αφ(x)| <∞.
The ‖φ‖α,γ are seminorms. They determine the topology of S.
C∞0 (Rn) ⊆ S
Easy to see that D j(φ)(ξ) = ξj φ(ξ) and xjφ(x)(ξ) = −D j φ(ξ).
More generally ξαDγφ(ξ) = Dα(−x)γφ(x)(ξ).
φ ∈ S =⇒ φ ∈ S (smoothness =⇒ decay, decay =⇒ smoothness)
Fourier inversion formula: φ(x) =∫φ(ξ)e2πiξx dξ.
Can also write asφ(−x) = φ(x).
We first show its validity for φ ∈ S.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 25 / 36
![Page 124: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/124.jpg)
Schwartz functions on Rn
Lp(Rn) spaces are not nested.
Not clear how to define f for f ∈ L2.
Schwartz class S: those φ ∈ C∞(Rn) s.t. for all multiindices α, γ
‖φ‖α,γ := supx∈Rn
|xγ∂αφ(x)| <∞.
The ‖φ‖α,γ are seminorms. They determine the topology of S.
C∞0 (Rn) ⊆ S
Easy to see that D j(φ)(ξ) = ξj φ(ξ) and xjφ(x)(ξ) = −D j φ(ξ).
More generally ξαDγφ(ξ) = Dα(−x)γφ(x)(ξ).
φ ∈ S =⇒ φ ∈ S (smoothness =⇒ decay, decay =⇒ smoothness)
Fourier inversion formula: φ(x) =∫φ(ξ)e2πiξx dξ.
Can also write asφ(−x) = φ(x).
We first show its validity for φ ∈ S.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 25 / 36
![Page 125: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/125.jpg)
Schwartz functions on Rn
Lp(Rn) spaces are not nested.
Not clear how to define f for f ∈ L2.
Schwartz class S: those φ ∈ C∞(Rn) s.t. for all multiindices α, γ
‖φ‖α,γ := supx∈Rn
|xγ∂αφ(x)| <∞.
The ‖φ‖α,γ are seminorms. They determine the topology of S.
C∞0 (Rn) ⊆ S
Easy to see that D j(φ)(ξ) = ξj φ(ξ) and xjφ(x)(ξ) = −D j φ(ξ).
More generally ξαDγφ(ξ) = Dα(−x)γφ(x)(ξ).
φ ∈ S =⇒ φ ∈ S (smoothness =⇒ decay, decay =⇒ smoothness)
Fourier inversion formula: φ(x) =∫φ(ξ)e2πiξx dξ.
Can also write asφ(−x) = φ(x).
We first show its validity for φ ∈ S.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 25 / 36
![Page 126: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/126.jpg)
Schwartz functions on Rn
Lp(Rn) spaces are not nested.
Not clear how to define f for f ∈ L2.
Schwartz class S: those φ ∈ C∞(Rn) s.t. for all multiindices α, γ
‖φ‖α,γ := supx∈Rn
|xγ∂αφ(x)| <∞.
The ‖φ‖α,γ are seminorms. They determine the topology of S.
C∞0 (Rn) ⊆ S
Easy to see that D j(φ)(ξ) = ξj φ(ξ) and xjφ(x)(ξ) = −D j φ(ξ).
More generally ξαDγφ(ξ) = Dα(−x)γφ(x)(ξ).
φ ∈ S =⇒ φ ∈ S (smoothness =⇒ decay, decay =⇒ smoothness)
Fourier inversion formula: φ(x) =∫φ(ξ)e2πiξx dξ.
Can also write asφ(−x) = φ(x).
We first show its validity for φ ∈ S.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 25 / 36
![Page 127: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/127.jpg)
Schwartz functions on Rn
Lp(Rn) spaces are not nested.
Not clear how to define f for f ∈ L2.
Schwartz class S: those φ ∈ C∞(Rn) s.t. for all multiindices α, γ
‖φ‖α,γ := supx∈Rn
|xγ∂αφ(x)| <∞.
The ‖φ‖α,γ are seminorms. They determine the topology of S.
C∞0 (Rn) ⊆ S
Easy to see that D j(φ)(ξ) = ξj φ(ξ) and xjφ(x)(ξ) = −D j φ(ξ).
More generally ξαDγφ(ξ) = Dα(−x)γφ(x)(ξ).
φ ∈ S =⇒ φ ∈ S (smoothness =⇒ decay, decay =⇒ smoothness)
Fourier inversion formula: φ(x) =∫φ(ξ)e2πiξx dξ.
Can also write asφ(−x) = φ(x).
We first show its validity for φ ∈ S.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 25 / 36
![Page 128: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/128.jpg)
Schwartz functions on Rn
Lp(Rn) spaces are not nested.
Not clear how to define f for f ∈ L2.
Schwartz class S: those φ ∈ C∞(Rn) s.t. for all multiindices α, γ
‖φ‖α,γ := supx∈Rn
|xγ∂αφ(x)| <∞.
The ‖φ‖α,γ are seminorms. They determine the topology of S.
C∞0 (Rn) ⊆ S
Easy to see that D j(φ)(ξ) = ξj φ(ξ) and xjφ(x)(ξ) = −D j φ(ξ).
More generally ξαDγφ(ξ) = Dα(−x)γφ(x)(ξ).
φ ∈ S =⇒ φ ∈ S (smoothness =⇒ decay, decay =⇒ smoothness)
Fourier inversion formula: φ(x) =∫φ(ξ)e2πiξx dξ.
Can also write asφ(−x) = φ(x).
We first show its validity for φ ∈ S.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 25 / 36
![Page 129: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/129.jpg)
Schwartz functions on Rn
Lp(Rn) spaces are not nested.
Not clear how to define f for f ∈ L2.
Schwartz class S: those φ ∈ C∞(Rn) s.t. for all multiindices α, γ
‖φ‖α,γ := supx∈Rn
|xγ∂αφ(x)| <∞.
The ‖φ‖α,γ are seminorms. They determine the topology of S.
C∞0 (Rn) ⊆ S
Easy to see that D j(φ)(ξ) = ξj φ(ξ) and xjφ(x)(ξ) = −D j φ(ξ).
More generally ξαDγφ(ξ) = Dα(−x)γφ(x)(ξ).
φ ∈ S =⇒ φ ∈ S (smoothness =⇒ decay, decay =⇒ smoothness)
Fourier inversion formula: φ(x) =∫φ(ξ)e2πiξx dξ.
Can also write asφ(−x) = φ(x).
We first show its validity for φ ∈ S.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 25 / 36
![Page 130: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/130.jpg)
Schwartz functions on Rn
Lp(Rn) spaces are not nested.
Not clear how to define f for f ∈ L2.
Schwartz class S: those φ ∈ C∞(Rn) s.t. for all multiindices α, γ
‖φ‖α,γ := supx∈Rn
|xγ∂αφ(x)| <∞.
The ‖φ‖α,γ are seminorms. They determine the topology of S.
C∞0 (Rn) ⊆ S
Easy to see that D j(φ)(ξ) = ξj φ(ξ) and xjφ(x)(ξ) = −D j φ(ξ).
More generally ξαDγφ(ξ) = Dα(−x)γφ(x)(ξ).
φ ∈ S =⇒ φ ∈ S (smoothness =⇒ decay, decay =⇒ smoothness)
Fourier inversion formula: φ(x) =∫φ(ξ)e2πiξx dξ.
Can also write asφ(−x) = φ(x).
We first show its validity for φ ∈ S.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 25 / 36
![Page 131: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/131.jpg)
Schwartz functions on Rn
Lp(Rn) spaces are not nested.
Not clear how to define f for f ∈ L2.
Schwartz class S: those φ ∈ C∞(Rn) s.t. for all multiindices α, γ
‖φ‖α,γ := supx∈Rn
|xγ∂αφ(x)| <∞.
The ‖φ‖α,γ are seminorms. They determine the topology of S.
C∞0 (Rn) ⊆ S
Easy to see that D j(φ)(ξ) = ξj φ(ξ) and xjφ(x)(ξ) = −D j φ(ξ).
More generally ξαDγφ(ξ) = Dα(−x)γφ(x)(ξ).
φ ∈ S =⇒ φ ∈ S (smoothness =⇒ decay, decay =⇒ smoothness)
Fourier inversion formula: φ(x) =∫φ(ξ)e2πiξx dξ.
Can also write asφ(−x) = φ(x).
We first show its validity for φ ∈ S.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 25 / 36
![Page 132: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/132.jpg)
Fourier inversion formula on S
shifting: f , g ∈ L1(Rn) =⇒∫
f g =∫
f g (Fubini)
Define the Gaussian function g(x) = (2π)−n/2e−|x |2/2, x ∈ Rn.
This normalization gives∫
g(x) =∫|x |2g(x) = 1.
Using Cauchy’s integral formula for analytic functions we proveg(ξ) = (2π)n/2g(2πξ). The Fourier inversion formula holds.
Write gε(x) = ε−ng(x/ε), an approximate identity.
We have gε(ξ) = (2π)n/2g(2πεξ), limε→0 gε(ξ) = 1.φ(−x) =
∫e2πiξx φ(ξ) dξ = limε→0
∫e2πiξx φ(ξ)gε(ξ) dξ (dom. conv.)
= limε→0
∫ φ(·+ x)(ξ)gε(ξ) dξ (FT of translation)
= limε→0
∫φ(x + y) gε(y) dy ( shifting)
= limε→0
∫φ(x + y)gε(−y) dy (FT inversion for gε).
= φ(x) (gε is an approximate identity)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 26 / 36
![Page 133: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/133.jpg)
Fourier inversion formula on S
shifting: f , g ∈ L1(Rn) =⇒∫
f g =∫
f g (Fubini)
Define the Gaussian function g(x) = (2π)−n/2e−|x |2/2, x ∈ Rn.
This normalization gives∫
g(x) =∫|x |2g(x) = 1.
Using Cauchy’s integral formula for analytic functions we proveg(ξ) = (2π)n/2g(2πξ). The Fourier inversion formula holds.
Write gε(x) = ε−ng(x/ε), an approximate identity.
We have gε(ξ) = (2π)n/2g(2πεξ), limε→0 gε(ξ) = 1.φ(−x) =
∫e2πiξx φ(ξ) dξ = limε→0
∫e2πiξx φ(ξ)gε(ξ) dξ (dom. conv.)
= limε→0
∫ φ(·+ x)(ξ)gε(ξ) dξ (FT of translation)
= limε→0
∫φ(x + y) gε(y) dy ( shifting)
= limε→0
∫φ(x + y)gε(−y) dy (FT inversion for gε).
= φ(x) (gε is an approximate identity)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 26 / 36
![Page 134: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/134.jpg)
Fourier inversion formula on S
shifting: f , g ∈ L1(Rn) =⇒∫
f g =∫
f g (Fubini)
Define the Gaussian function g(x) = (2π)−n/2e−|x |2/2, x ∈ Rn.
This normalization gives∫
g(x) =∫|x |2g(x) = 1.
Using Cauchy’s integral formula for analytic functions we proveg(ξ) = (2π)n/2g(2πξ). The Fourier inversion formula holds.
Write gε(x) = ε−ng(x/ε), an approximate identity.
We have gε(ξ) = (2π)n/2g(2πεξ), limε→0 gε(ξ) = 1.φ(−x) =
∫e2πiξx φ(ξ) dξ = limε→0
∫e2πiξx φ(ξ)gε(ξ) dξ (dom. conv.)
= limε→0
∫ φ(·+ x)(ξ)gε(ξ) dξ (FT of translation)
= limε→0
∫φ(x + y) gε(y) dy ( shifting)
= limε→0
∫φ(x + y)gε(−y) dy (FT inversion for gε).
= φ(x) (gε is an approximate identity)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 26 / 36
![Page 135: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/135.jpg)
Fourier inversion formula on S
shifting: f , g ∈ L1(Rn) =⇒∫
f g =∫
f g (Fubini)
Define the Gaussian function g(x) = (2π)−n/2e−|x |2/2, x ∈ Rn.
This normalization gives∫
g(x) =∫|x |2g(x) = 1.
Using Cauchy’s integral formula for analytic functions we proveg(ξ) = (2π)n/2g(2πξ). The Fourier inversion formula holds.
Write gε(x) = ε−ng(x/ε), an approximate identity.
We have gε(ξ) = (2π)n/2g(2πεξ), limε→0 gε(ξ) = 1.φ(−x) =
∫e2πiξx φ(ξ) dξ = limε→0
∫e2πiξx φ(ξ)gε(ξ) dξ (dom. conv.)
= limε→0
∫ φ(·+ x)(ξ)gε(ξ) dξ (FT of translation)
= limε→0
∫φ(x + y) gε(y) dy ( shifting)
= limε→0
∫φ(x + y)gε(−y) dy (FT inversion for gε).
= φ(x) (gε is an approximate identity)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 26 / 36
![Page 136: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/136.jpg)
Fourier inversion formula on S
shifting: f , g ∈ L1(Rn) =⇒∫
f g =∫
f g (Fubini)
Define the Gaussian function g(x) = (2π)−n/2e−|x |2/2, x ∈ Rn.
This normalization gives∫
g(x) =∫|x |2g(x) = 1.
Using Cauchy’s integral formula for analytic functions we proveg(ξ) = (2π)n/2g(2πξ). The Fourier inversion formula holds.
Write gε(x) = ε−ng(x/ε), an approximate identity.
We have gε(ξ) = (2π)n/2g(2πεξ), limε→0 gε(ξ) = 1.
φ(−x) =
∫e2πiξx φ(ξ) dξ = limε→0
∫e2πiξx φ(ξ)gε(ξ) dξ (dom. conv.)
= limε→0
∫ φ(·+ x)(ξ)gε(ξ) dξ (FT of translation)
= limε→0
∫φ(x + y) gε(y) dy ( shifting)
= limε→0
∫φ(x + y)gε(−y) dy (FT inversion for gε).
= φ(x) (gε is an approximate identity)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 26 / 36
![Page 137: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/137.jpg)
Fourier inversion formula on S
shifting: f , g ∈ L1(Rn) =⇒∫
f g =∫
f g (Fubini)
Define the Gaussian function g(x) = (2π)−n/2e−|x |2/2, x ∈ Rn.
This normalization gives∫
g(x) =∫|x |2g(x) = 1.
Using Cauchy’s integral formula for analytic functions we proveg(ξ) = (2π)n/2g(2πξ). The Fourier inversion formula holds.
Write gε(x) = ε−ng(x/ε), an approximate identity.
We have gε(ξ) = (2π)n/2g(2πεξ), limε→0 gε(ξ) = 1.φ(−x) =
∫e2πiξx φ(ξ) dξ = limε→0
∫e2πiξx φ(ξ)gε(ξ) dξ (dom. conv.)
= limε→0
∫ φ(·+ x)(ξ)gε(ξ) dξ (FT of translation)
= limε→0
∫φ(x + y) gε(y) dy ( shifting)
= limε→0
∫φ(x + y)gε(−y) dy (FT inversion for gε).
= φ(x) (gε is an approximate identity)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 26 / 36
![Page 138: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/138.jpg)
Fourier inversion formula on S
shifting: f , g ∈ L1(Rn) =⇒∫
f g =∫
f g (Fubini)
Define the Gaussian function g(x) = (2π)−n/2e−|x |2/2, x ∈ Rn.
This normalization gives∫
g(x) =∫|x |2g(x) = 1.
Using Cauchy’s integral formula for analytic functions we proveg(ξ) = (2π)n/2g(2πξ). The Fourier inversion formula holds.
Write gε(x) = ε−ng(x/ε), an approximate identity.
We have gε(ξ) = (2π)n/2g(2πεξ), limε→0 gε(ξ) = 1.φ(−x) =
∫e2πiξx φ(ξ) dξ = limε→0
∫e2πiξx φ(ξ)gε(ξ) dξ (dom. conv.)
= limε→0
∫ φ(·+ x)(ξ)gε(ξ) dξ (FT of translation)
= limε→0
∫φ(x + y) gε(y) dy ( shifting)
= limε→0
∫φ(x + y)gε(−y) dy (FT inversion for gε).
= φ(x) (gε is an approximate identity)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 26 / 36
![Page 139: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/139.jpg)
Fourier inversion formula on S
shifting: f , g ∈ L1(Rn) =⇒∫
f g =∫
f g (Fubini)
Define the Gaussian function g(x) = (2π)−n/2e−|x |2/2, x ∈ Rn.
This normalization gives∫
g(x) =∫|x |2g(x) = 1.
Using Cauchy’s integral formula for analytic functions we proveg(ξ) = (2π)n/2g(2πξ). The Fourier inversion formula holds.
Write gε(x) = ε−ng(x/ε), an approximate identity.
We have gε(ξ) = (2π)n/2g(2πεξ), limε→0 gε(ξ) = 1.φ(−x) =
∫e2πiξx φ(ξ) dξ = limε→0
∫e2πiξx φ(ξ)gε(ξ) dξ (dom. conv.)
= limε→0
∫ φ(·+ x)(ξ)gε(ξ) dξ (FT of translation)
= limε→0
∫φ(x + y) gε(y) dy ( shifting)
= limε→0
∫φ(x + y)gε(−y) dy (FT inversion for gε).
= φ(x) (gε is an approximate identity)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 26 / 36
![Page 140: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/140.jpg)
Fourier inversion formula on S
shifting: f , g ∈ L1(Rn) =⇒∫
f g =∫
f g (Fubini)
Define the Gaussian function g(x) = (2π)−n/2e−|x |2/2, x ∈ Rn.
This normalization gives∫
g(x) =∫|x |2g(x) = 1.
Using Cauchy’s integral formula for analytic functions we proveg(ξ) = (2π)n/2g(2πξ). The Fourier inversion formula holds.
Write gε(x) = ε−ng(x/ε), an approximate identity.
We have gε(ξ) = (2π)n/2g(2πεξ), limε→0 gε(ξ) = 1.φ(−x) =
∫e2πiξx φ(ξ) dξ = limε→0
∫e2πiξx φ(ξ)gε(ξ) dξ (dom. conv.)
= limε→0
∫ φ(·+ x)(ξ)gε(ξ) dξ (FT of translation)
= limε→0
∫φ(x + y) gε(y) dy ( shifting)
= limε→0
∫φ(x + y)gε(−y) dy (FT inversion for gε).
= φ(x) (gε is an approximate identity)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 26 / 36
![Page 141: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/141.jpg)
Fourier inversion formula on S
shifting: f , g ∈ L1(Rn) =⇒∫
f g =∫
f g (Fubini)
Define the Gaussian function g(x) = (2π)−n/2e−|x |2/2, x ∈ Rn.
This normalization gives∫
g(x) =∫|x |2g(x) = 1.
Using Cauchy’s integral formula for analytic functions we proveg(ξ) = (2π)n/2g(2πξ). The Fourier inversion formula holds.
Write gε(x) = ε−ng(x/ε), an approximate identity.
We have gε(ξ) = (2π)n/2g(2πεξ), limε→0 gε(ξ) = 1.φ(−x) =
∫e2πiξx φ(ξ) dξ = limε→0
∫e2πiξx φ(ξ)gε(ξ) dξ (dom. conv.)
= limε→0
∫ φ(·+ x)(ξ)gε(ξ) dξ (FT of translation)
= limε→0
∫φ(x + y) gε(y) dy ( shifting)
= limε→0
∫φ(x + y)gε(−y) dy (FT inversion for gε).
= φ(x) (gε is an approximate identity)
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 26 / 36
![Page 142: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/142.jpg)
FT on L2(Rn)
Preservation of inner product:∫φψ =
∫φψ, for φ, ψ ∈ S
Fourier inversion impliesψ = ψ. Use shifting.
Parseval: f ∈ S: ‖f ‖2 =∥∥∥f
∥∥∥2.
S dense in L2(Rn): FT extends to L2 and f → f is an isometry on L2.
By interpolation FT is defined on Lp, 1 ≤ p ≤ 2, and satisfies theHausdorff-Young inequality:∥∥∥f
∥∥∥q≤ Cp‖f ‖p,
1
p+
1
q= 1.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 27 / 36
![Page 143: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/143.jpg)
FT on L2(Rn)
Preservation of inner product:∫φψ =
∫φψ, for φ, ψ ∈ S
Fourier inversion impliesψ = ψ. Use shifting.
Parseval: f ∈ S: ‖f ‖2 =∥∥∥f
∥∥∥2.
S dense in L2(Rn): FT extends to L2 and f → f is an isometry on L2.
By interpolation FT is defined on Lp, 1 ≤ p ≤ 2, and satisfies theHausdorff-Young inequality:∥∥∥f
∥∥∥q≤ Cp‖f ‖p,
1
p+
1
q= 1.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 27 / 36
![Page 144: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/144.jpg)
FT on L2(Rn)
Preservation of inner product:∫φψ =
∫φψ, for φ, ψ ∈ S
Fourier inversion impliesψ = ψ. Use shifting.
Parseval: f ∈ S: ‖f ‖2 =∥∥∥f
∥∥∥2.
S dense in L2(Rn): FT extends to L2 and f → f is an isometry on L2.
By interpolation FT is defined on Lp, 1 ≤ p ≤ 2, and satisfies theHausdorff-Young inequality:∥∥∥f
∥∥∥q≤ Cp‖f ‖p,
1
p+
1
q= 1.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 27 / 36
![Page 145: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/145.jpg)
FT on L2(Rn)
Preservation of inner product:∫φψ =
∫φψ, for φ, ψ ∈ S
Fourier inversion impliesψ = ψ. Use shifting.
Parseval: f ∈ S: ‖f ‖2 =∥∥∥f
∥∥∥2.
S dense in L2(Rn): FT extends to L2 and f → f is an isometry on L2.
By interpolation FT is defined on Lp, 1 ≤ p ≤ 2, and satisfies theHausdorff-Young inequality:∥∥∥f
∥∥∥q≤ Cp‖f ‖p,
1
p+
1
q= 1.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 27 / 36
![Page 146: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/146.jpg)
Tempered distributions
Tempered distributions: S ′ is the space of continuous linearfunctionals on S.
FT defined on S ′: for u ∈ S ′ we define u(φ) = u(φ), for φ ∈ S.
Fourier inversion for S ′: u(φ(x)) = u(φ(−x)).
u → u is an isomorphism on S ′
1 ≤ p ≤ ∞ : Lp ⊆ S ′. If f ∈ Lp this mapping is in S ′:
φ→∫
f φ
Also S ⊆ S ′.Tempered measures:
∫(1 + |x |)−k d |µ|(x) <∞, for some k ∈ N.
These are in S ′.Differentiation defined as: (∂αu)(φ) = (−1)|α|u(∂αφ).
FT of Lp functions or tempered measures defined in S ′
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 28 / 36
![Page 147: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/147.jpg)
Tempered distributions
Tempered distributions: S ′ is the space of continuous linearfunctionals on S.
FT defined on S ′: for u ∈ S ′ we define u(φ) = u(φ), for φ ∈ S.
Fourier inversion for S ′: u(φ(x)) = u(φ(−x)).
u → u is an isomorphism on S ′
1 ≤ p ≤ ∞ : Lp ⊆ S ′. If f ∈ Lp this mapping is in S ′:
φ→∫
f φ
Also S ⊆ S ′.Tempered measures:
∫(1 + |x |)−k d |µ|(x) <∞, for some k ∈ N.
These are in S ′.Differentiation defined as: (∂αu)(φ) = (−1)|α|u(∂αφ).
FT of Lp functions or tempered measures defined in S ′
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 28 / 36
![Page 148: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/148.jpg)
Tempered distributions
Tempered distributions: S ′ is the space of continuous linearfunctionals on S.
FT defined on S ′: for u ∈ S ′ we define u(φ) = u(φ), for φ ∈ S.
Fourier inversion for S ′: u(φ(x)) = u(φ(−x)).
u → u is an isomorphism on S ′
1 ≤ p ≤ ∞ : Lp ⊆ S ′. If f ∈ Lp this mapping is in S ′:
φ→∫
f φ
Also S ⊆ S ′.Tempered measures:
∫(1 + |x |)−k d |µ|(x) <∞, for some k ∈ N.
These are in S ′.Differentiation defined as: (∂αu)(φ) = (−1)|α|u(∂αφ).
FT of Lp functions or tempered measures defined in S ′
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 28 / 36
![Page 149: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/149.jpg)
Tempered distributions
Tempered distributions: S ′ is the space of continuous linearfunctionals on S.
FT defined on S ′: for u ∈ S ′ we define u(φ) = u(φ), for φ ∈ S.
Fourier inversion for S ′: u(φ(x)) = u(φ(−x)).
u → u is an isomorphism on S ′
1 ≤ p ≤ ∞ : Lp ⊆ S ′. If f ∈ Lp this mapping is in S ′:
φ→∫
f φ
Also S ⊆ S ′.Tempered measures:
∫(1 + |x |)−k d |µ|(x) <∞, for some k ∈ N.
These are in S ′.Differentiation defined as: (∂αu)(φ) = (−1)|α|u(∂αφ).
FT of Lp functions or tempered measures defined in S ′
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 28 / 36
![Page 150: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/150.jpg)
Tempered distributions
Tempered distributions: S ′ is the space of continuous linearfunctionals on S.
FT defined on S ′: for u ∈ S ′ we define u(φ) = u(φ), for φ ∈ S.
Fourier inversion for S ′: u(φ(x)) = u(φ(−x)).
u → u is an isomorphism on S ′
1 ≤ p ≤ ∞ : Lp ⊆ S ′. If f ∈ Lp this mapping is in S ′:
φ→∫
f φ
Also S ⊆ S ′.
Tempered measures:∫
(1 + |x |)−k d |µ|(x) <∞, for some k ∈ N.These are in S ′.Differentiation defined as: (∂αu)(φ) = (−1)|α|u(∂αφ).
FT of Lp functions or tempered measures defined in S ′
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 28 / 36
![Page 151: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/151.jpg)
Tempered distributions
Tempered distributions: S ′ is the space of continuous linearfunctionals on S.
FT defined on S ′: for u ∈ S ′ we define u(φ) = u(φ), for φ ∈ S.
Fourier inversion for S ′: u(φ(x)) = u(φ(−x)).
u → u is an isomorphism on S ′
1 ≤ p ≤ ∞ : Lp ⊆ S ′. If f ∈ Lp this mapping is in S ′:
φ→∫
f φ
Also S ⊆ S ′.Tempered measures:
∫(1 + |x |)−k d |µ|(x) <∞, for some k ∈ N.
These are in S ′.
Differentiation defined as: (∂αu)(φ) = (−1)|α|u(∂αφ).
FT of Lp functions or tempered measures defined in S ′
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 28 / 36
![Page 152: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/152.jpg)
Tempered distributions
Tempered distributions: S ′ is the space of continuous linearfunctionals on S.
FT defined on S ′: for u ∈ S ′ we define u(φ) = u(φ), for φ ∈ S.
Fourier inversion for S ′: u(φ(x)) = u(φ(−x)).
u → u is an isomorphism on S ′
1 ≤ p ≤ ∞ : Lp ⊆ S ′. If f ∈ Lp this mapping is in S ′:
φ→∫
f φ
Also S ⊆ S ′.Tempered measures:
∫(1 + |x |)−k d |µ|(x) <∞, for some k ∈ N.
These are in S ′.Differentiation defined as: (∂αu)(φ) = (−1)|α|u(∂αφ).
FT of Lp functions or tempered measures defined in S ′
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 28 / 36
![Page 153: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/153.jpg)
Tempered distributions
Tempered distributions: S ′ is the space of continuous linearfunctionals on S.
FT defined on S ′: for u ∈ S ′ we define u(φ) = u(φ), for φ ∈ S.
Fourier inversion for S ′: u(φ(x)) = u(φ(−x)).
u → u is an isomorphism on S ′
1 ≤ p ≤ ∞ : Lp ⊆ S ′. If f ∈ Lp this mapping is in S ′:
φ→∫
f φ
Also S ⊆ S ′.Tempered measures:
∫(1 + |x |)−k d |µ|(x) <∞, for some k ∈ N.
These are in S ′.Differentiation defined as: (∂αu)(φ) = (−1)|α|u(∂αφ).
FT of Lp functions or tempered measures defined in S ′
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 28 / 36
![Page 154: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/154.jpg)
Examples of tempered distributions and their FT
u = δ0, u = 1
u = Dαδ0. To find its FT
Dαδ0(φ) = (Dαδ0)(φ) = (−1)|α|δ0(Dαφ) = (−1)|α|δ0( (−x)αφ(x))
= (−1)|α|δ0((−x)αφ(x)) =
∫xαφ(x)
So Dαδ0 = xα.
u = xα, u = Dαδ0
u =∑J
j=1 ajδpj , u(ξ) =∑J
j=1 aje2πipjξ.
Poisson Summation Formula (PSF): u =∑
k∈Zn δk , u = u
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 29 / 36
![Page 155: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/155.jpg)
Examples of tempered distributions and their FT
u = δ0, u = 1
u = Dαδ0. To find its FT
Dαδ0(φ) = (Dαδ0)(φ) = (−1)|α|δ0(Dαφ) = (−1)|α|δ0( (−x)αφ(x))
= (−1)|α|δ0((−x)αφ(x)) =
∫xαφ(x)
So Dαδ0 = xα.
u = xα, u = Dαδ0
u =∑J
j=1 ajδpj , u(ξ) =∑J
j=1 aje2πipjξ.
Poisson Summation Formula (PSF): u =∑
k∈Zn δk , u = u
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 29 / 36
![Page 156: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/156.jpg)
Examples of tempered distributions and their FT
u = δ0, u = 1
u = Dαδ0. To find its FT
Dαδ0(φ) = (Dαδ0)(φ) = (−1)|α|δ0(Dαφ) = (−1)|α|δ0( (−x)αφ(x))
= (−1)|α|δ0((−x)αφ(x)) =
∫xαφ(x)
So Dαδ0 = xα.
u = xα, u = Dαδ0
u =∑J
j=1 ajδpj , u(ξ) =∑J
j=1 aje2πipjξ.
Poisson Summation Formula (PSF): u =∑
k∈Zn δk , u = u
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 29 / 36
![Page 157: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/157.jpg)
Examples of tempered distributions and their FT
u = δ0, u = 1
u = Dαδ0. To find its FT
Dαδ0(φ) = (Dαδ0)(φ) = (−1)|α|δ0(Dαφ) = (−1)|α|δ0( (−x)αφ(x))
= (−1)|α|δ0((−x)αφ(x)) =
∫xαφ(x)
So Dαδ0 = xα.
u = xα, u = Dαδ0
u =∑J
j=1 ajδpj , u(ξ) =∑J
j=1 aje2πipjξ.
Poisson Summation Formula (PSF): u =∑
k∈Zn δk , u = u
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 29 / 36
![Page 158: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/158.jpg)
Examples of tempered distributions and their FT
u = δ0, u = 1
u = Dαδ0. To find its FT
Dαδ0(φ) = (Dαδ0)(φ) = (−1)|α|δ0(Dαφ) = (−1)|α|δ0( (−x)αφ(x))
= (−1)|α|δ0((−x)αφ(x)) =
∫xαφ(x)
So Dαδ0 = xα.
u = xα, u = Dαδ0
u =∑J
j=1 ajδpj , u(ξ) =∑J
j=1 aje2πipjξ.
Poisson Summation Formula (PSF): u =∑
k∈Zn δk , u = u
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 29 / 36
![Page 159: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/159.jpg)
Examples of tempered distributions and their FT
u = δ0, u = 1
u = Dαδ0. To find its FT
Dαδ0(φ) = (Dαδ0)(φ) = (−1)|α|δ0(Dαφ) = (−1)|α|δ0( (−x)αφ(x))
= (−1)|α|δ0((−x)αφ(x)) =
∫xαφ(x)
So Dαδ0 = xα.
u = xα, u = Dαδ0
u =∑J
j=1 ajδpj , u(ξ) =∑J
j=1 aje2πipjξ.
Poisson Summation Formula (PSF): u =∑
k∈Zn δk , u = u
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 29 / 36
![Page 160: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/160.jpg)
Proof of the Poisson Summation Formula
φ ∈ S. Define g(x) =∑
k∈Zn φ(x + k).
g has Zn as a period lattice: g(x + k) = g(x), x ∈ Rn, k ∈ Zn.
The periodization g may be viewed as g : Tn → C.
FT of g lives on Tn = Zn. The Fourier coefficients are
g(k) =
∫Tn
∑m∈Zn
φ(x + m)e−2πikx dx = φ(k).
From decay of φ follows that the FS of g(x) converges absolutely anduniformly and
g(x) =∑k∈Zn
φ(k)e2πikx .
x = 0 gives the PSF:∑
k∈Zn φ(k) =∑
k∈Zn φ(k).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 30 / 36
![Page 161: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/161.jpg)
Proof of the Poisson Summation Formula
φ ∈ S. Define g(x) =∑
k∈Zn φ(x + k).
g has Zn as a period lattice: g(x + k) = g(x), x ∈ Rn, k ∈ Zn.
The periodization g may be viewed as g : Tn → C.
FT of g lives on Tn = Zn. The Fourier coefficients are
g(k) =
∫Tn
∑m∈Zn
φ(x + m)e−2πikx dx = φ(k).
From decay of φ follows that the FS of g(x) converges absolutely anduniformly and
g(x) =∑k∈Zn
φ(k)e2πikx .
x = 0 gives the PSF:∑
k∈Zn φ(k) =∑
k∈Zn φ(k).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 30 / 36
![Page 162: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/162.jpg)
Proof of the Poisson Summation Formula
φ ∈ S. Define g(x) =∑
k∈Zn φ(x + k).
g has Zn as a period lattice: g(x + k) = g(x), x ∈ Rn, k ∈ Zn.
The periodization g may be viewed as g : Tn → C.
FT of g lives on Tn = Zn. The Fourier coefficients are
g(k) =
∫Tn
∑m∈Zn
φ(x + m)e−2πikx dx = φ(k).
From decay of φ follows that the FS of g(x) converges absolutely anduniformly and
g(x) =∑k∈Zn
φ(k)e2πikx .
x = 0 gives the PSF:∑
k∈Zn φ(k) =∑
k∈Zn φ(k).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 30 / 36
![Page 163: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/163.jpg)
Proof of the Poisson Summation Formula
φ ∈ S. Define g(x) =∑
k∈Zn φ(x + k).
g has Zn as a period lattice: g(x + k) = g(x), x ∈ Rn, k ∈ Zn.
The periodization g may be viewed as g : Tn → C.
FT of g lives on Tn = Zn. The Fourier coefficients are
g(k) =
∫Tn
∑m∈Zn
φ(x + m)e−2πikx dx = φ(k).
From decay of φ follows that the FS of g(x) converges absolutely anduniformly and
g(x) =∑k∈Zn
φ(k)e2πikx .
x = 0 gives the PSF:∑
k∈Zn φ(k) =∑
k∈Zn φ(k).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 30 / 36
![Page 164: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/164.jpg)
Proof of the Poisson Summation Formula
φ ∈ S. Define g(x) =∑
k∈Zn φ(x + k).
g has Zn as a period lattice: g(x + k) = g(x), x ∈ Rn, k ∈ Zn.
The periodization g may be viewed as g : Tn → C.
FT of g lives on Tn = Zn. The Fourier coefficients are
g(k) =
∫Tn
∑m∈Zn
φ(x + m)e−2πikx dx = φ(k).
From decay of φ follows that the FS of g(x) converges absolutely anduniformly and
g(x) =∑k∈Zn
φ(k)e2πikx .
x = 0 gives the PSF:∑
k∈Zn φ(k) =∑
k∈Zn φ(k).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 30 / 36
![Page 165: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/165.jpg)
Proof of the Poisson Summation Formula
φ ∈ S. Define g(x) =∑
k∈Zn φ(x + k).
g has Zn as a period lattice: g(x + k) = g(x), x ∈ Rn, k ∈ Zn.
The periodization g may be viewed as g : Tn → C.
FT of g lives on Tn = Zn. The Fourier coefficients are
g(k) =
∫Tn
∑m∈Zn
φ(x + m)e−2πikx dx = φ(k).
From decay of φ follows that the FS of g(x) converges absolutely anduniformly and
g(x) =∑k∈Zn
φ(k)e2πikx .
x = 0 gives the PSF:∑
k∈Zn φ(k) =∑
k∈Zn φ(k).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 30 / 36
![Page 166: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/166.jpg)
FT behavior under linear transformation
Suppose T : Rn → Rn is a non-singular linear operator. u ∈ S ′,v = u ◦ T .
Change of variables formula for integration implies
v =1
|det T |u ◦ T−>.
Write Rn = H ⊕ H⊥, H a linear subspace.
Projection onto subspace defined by
(πH f )(h) :=
∫H>
f (h + x) dx , (h ∈ H).
For ξ ∈ H: πH f (ξ) = f (ξ) (Fubini).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 31 / 36
![Page 167: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/167.jpg)
FT behavior under linear transformation
Suppose T : Rn → Rn is a non-singular linear operator. u ∈ S ′,v = u ◦ T .
Change of variables formula for integration implies
v =1
|det T |u ◦ T−>.
Write Rn = H ⊕ H⊥, H a linear subspace.
Projection onto subspace defined by
(πH f )(h) :=
∫H>
f (h + x) dx , (h ∈ H).
For ξ ∈ H: πH f (ξ) = f (ξ) (Fubini).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 31 / 36
![Page 168: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/168.jpg)
FT behavior under linear transformation
Suppose T : Rn → Rn is a non-singular linear operator. u ∈ S ′,v = u ◦ T .
Change of variables formula for integration implies
v =1
|det T |u ◦ T−>.
Write Rn = H ⊕ H⊥, H a linear subspace.
Projection onto subspace defined by
(πH f )(h) :=
∫H>
f (h + x) dx , (h ∈ H).
For ξ ∈ H: πH f (ξ) = f (ξ) (Fubini).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 31 / 36
![Page 169: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/169.jpg)
FT behavior under linear transformation
Suppose T : Rn → Rn is a non-singular linear operator. u ∈ S ′,v = u ◦ T .
Change of variables formula for integration implies
v =1
|det T |u ◦ T−>.
Write Rn = H ⊕ H⊥, H a linear subspace.
Projection onto subspace defined by
(πH f )(h) :=
∫H>
f (h + x) dx , (h ∈ H).
For ξ ∈ H: πH f (ξ) = f (ξ) (Fubini).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 31 / 36
![Page 170: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/170.jpg)
FT behavior under linear transformation
Suppose T : Rn → Rn is a non-singular linear operator. u ∈ S ′,v = u ◦ T .
Change of variables formula for integration implies
v =1
|det T |u ◦ T−>.
Write Rn = H ⊕ H⊥, H a linear subspace.
Projection onto subspace defined by
(πH f )(h) :=
∫H>
f (h + x) dx , (h ∈ H).
For ξ ∈ H: πH f (ξ) = f (ξ) (Fubini).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 31 / 36
![Page 171: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/171.jpg)
Analyticity of the FT
Compact support: f : R → C, f ∈ L1(R), f (x) = 0 for |x | > R.
FT defined by
f (ξ) =
∫R
f (x)e−2πiξx dx .
Allow ξ ∈ C, ξ = s + it in the formula.
f (s + it) =
∫f (x)e2πtxe−2πisx dx
Compact support implies f (x)e2πtx ∈ L1(R), so integral is defined.Since e−2πixξ is analytic for all ξ ∈ C, so is f (ξ).Paley–Wiener: f ∈ L2(R). The following are equivalent:(a) f is the restriction on R of a function F holomorphic in the strip{z : |=z | < a} which satisfies∫
|F (x + iy)|2 dx ≤ C , (|y | < a)
(b) ea|ξ|f (ξ) ∈ L2(R).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 32 / 36
![Page 172: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/172.jpg)
Analyticity of the FT
Compact support: f : R → C, f ∈ L1(R), f (x) = 0 for |x | > R.FT defined by
f (ξ) =
∫R
f (x)e−2πiξx dx .
Allow ξ ∈ C, ξ = s + it in the formula.
f (s + it) =
∫f (x)e2πtxe−2πisx dx
Compact support implies f (x)e2πtx ∈ L1(R), so integral is defined.Since e−2πixξ is analytic for all ξ ∈ C, so is f (ξ).Paley–Wiener: f ∈ L2(R). The following are equivalent:(a) f is the restriction on R of a function F holomorphic in the strip{z : |=z | < a} which satisfies∫
|F (x + iy)|2 dx ≤ C , (|y | < a)
(b) ea|ξ|f (ξ) ∈ L2(R).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 32 / 36
![Page 173: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/173.jpg)
Analyticity of the FT
Compact support: f : R → C, f ∈ L1(R), f (x) = 0 for |x | > R.FT defined by
f (ξ) =
∫R
f (x)e−2πiξx dx .
Allow ξ ∈ C, ξ = s + it in the formula.
f (s + it) =
∫f (x)e2πtxe−2πisx dx
Compact support implies f (x)e2πtx ∈ L1(R), so integral is defined.Since e−2πixξ is analytic for all ξ ∈ C, so is f (ξ).Paley–Wiener: f ∈ L2(R). The following are equivalent:(a) f is the restriction on R of a function F holomorphic in the strip{z : |=z | < a} which satisfies∫
|F (x + iy)|2 dx ≤ C , (|y | < a)
(b) ea|ξ|f (ξ) ∈ L2(R).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 32 / 36
![Page 174: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/174.jpg)
Analyticity of the FT
Compact support: f : R → C, f ∈ L1(R), f (x) = 0 for |x | > R.FT defined by
f (ξ) =
∫R
f (x)e−2πiξx dx .
Allow ξ ∈ C, ξ = s + it in the formula.
f (s + it) =
∫f (x)e2πtxe−2πisx dx
Compact support implies f (x)e2πtx ∈ L1(R), so integral is defined.
Since e−2πixξ is analytic for all ξ ∈ C, so is f (ξ).Paley–Wiener: f ∈ L2(R). The following are equivalent:(a) f is the restriction on R of a function F holomorphic in the strip{z : |=z | < a} which satisfies∫
|F (x + iy)|2 dx ≤ C , (|y | < a)
(b) ea|ξ|f (ξ) ∈ L2(R).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 32 / 36
![Page 175: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/175.jpg)
Analyticity of the FT
Compact support: f : R → C, f ∈ L1(R), f (x) = 0 for |x | > R.FT defined by
f (ξ) =
∫R
f (x)e−2πiξx dx .
Allow ξ ∈ C, ξ = s + it in the formula.
f (s + it) =
∫f (x)e2πtxe−2πisx dx
Compact support implies f (x)e2πtx ∈ L1(R), so integral is defined.Since e−2πixξ is analytic for all ξ ∈ C, so is f (ξ).
Paley–Wiener: f ∈ L2(R). The following are equivalent:(a) f is the restriction on R of a function F holomorphic in the strip{z : |=z | < a} which satisfies∫
|F (x + iy)|2 dx ≤ C , (|y | < a)
(b) ea|ξ|f (ξ) ∈ L2(R).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 32 / 36
![Page 176: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/176.jpg)
Analyticity of the FT
Compact support: f : R → C, f ∈ L1(R), f (x) = 0 for |x | > R.FT defined by
f (ξ) =
∫R
f (x)e−2πiξx dx .
Allow ξ ∈ C, ξ = s + it in the formula.
f (s + it) =
∫f (x)e2πtxe−2πisx dx
Compact support implies f (x)e2πtx ∈ L1(R), so integral is defined.Since e−2πixξ is analytic for all ξ ∈ C, so is f (ξ).Paley–Wiener: f ∈ L2(R). The following are equivalent:(a) f is the restriction on R of a function F holomorphic in the strip{z : |=z | < a} which satisfies∫
|F (x + iy)|2 dx ≤ C , (|y | < a)
(b) ea|ξ|f (ξ) ∈ L2(R).
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 32 / 36
![Page 177: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/177.jpg)
Application: the Steinhaus tiling problem
Question of Steinhaus: Is there E ⊆ R2 such that no matter howtranslated and rotated it always contains exactly one point withinteger coordinates?
Two versions: E is required to be measurable or not
Non-measurable version was answered in the affirmative by Jacksonand Mauldin a few years ago.
Measurable version remains open.
Equivalent form (Rθ is rotation by θ):∑k∈Z2
1RθE (t + k) = 1, (0 ≤ θ < 2π, t ∈ R2). (5)
We prove: there is no bounded measurable Steinhaus set.
Integrating (5) for t ∈ [0, 1]2 we obtain |E | = 1.
LHS of (5) is the Z2-periodization of 1RθE . Hence 1RθE (k) = 0,k ∈ Z2 \ {0}.1E (ξ) = 0, whenever ξ on a circle through a lattice point
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 33 / 36
![Page 178: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/178.jpg)
Application: the Steinhaus tiling problem
Question of Steinhaus: Is there E ⊆ R2 such that no matter howtranslated and rotated it always contains exactly one point withinteger coordinates?
Two versions: E is required to be measurable or not
Non-measurable version was answered in the affirmative by Jacksonand Mauldin a few years ago.
Measurable version remains open.
Equivalent form (Rθ is rotation by θ):∑k∈Z2
1RθE (t + k) = 1, (0 ≤ θ < 2π, t ∈ R2). (5)
We prove: there is no bounded measurable Steinhaus set.
Integrating (5) for t ∈ [0, 1]2 we obtain |E | = 1.
LHS of (5) is the Z2-periodization of 1RθE . Hence 1RθE (k) = 0,k ∈ Z2 \ {0}.1E (ξ) = 0, whenever ξ on a circle through a lattice point
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 33 / 36
![Page 179: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/179.jpg)
Application: the Steinhaus tiling problem
Question of Steinhaus: Is there E ⊆ R2 such that no matter howtranslated and rotated it always contains exactly one point withinteger coordinates?
Two versions: E is required to be measurable or not
Non-measurable version was answered in the affirmative by Jacksonand Mauldin a few years ago.
Measurable version remains open.
Equivalent form (Rθ is rotation by θ):∑k∈Z2
1RθE (t + k) = 1, (0 ≤ θ < 2π, t ∈ R2). (5)
We prove: there is no bounded measurable Steinhaus set.
Integrating (5) for t ∈ [0, 1]2 we obtain |E | = 1.
LHS of (5) is the Z2-periodization of 1RθE . Hence 1RθE (k) = 0,k ∈ Z2 \ {0}.1E (ξ) = 0, whenever ξ on a circle through a lattice point
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 33 / 36
![Page 180: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/180.jpg)
Application: the Steinhaus tiling problem
Question of Steinhaus: Is there E ⊆ R2 such that no matter howtranslated and rotated it always contains exactly one point withinteger coordinates?
Two versions: E is required to be measurable or not
Non-measurable version was answered in the affirmative by Jacksonand Mauldin a few years ago.
Measurable version remains open.
Equivalent form (Rθ is rotation by θ):∑k∈Z2
1RθE (t + k) = 1, (0 ≤ θ < 2π, t ∈ R2). (5)
We prove: there is no bounded measurable Steinhaus set.
Integrating (5) for t ∈ [0, 1]2 we obtain |E | = 1.
LHS of (5) is the Z2-periodization of 1RθE . Hence 1RθE (k) = 0,k ∈ Z2 \ {0}.1E (ξ) = 0, whenever ξ on a circle through a lattice point
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 33 / 36
![Page 181: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/181.jpg)
Application: the Steinhaus tiling problem
Question of Steinhaus: Is there E ⊆ R2 such that no matter howtranslated and rotated it always contains exactly one point withinteger coordinates?
Two versions: E is required to be measurable or not
Non-measurable version was answered in the affirmative by Jacksonand Mauldin a few years ago.
Measurable version remains open.
Equivalent form (Rθ is rotation by θ):∑k∈Z2
1RθE (t + k) = 1, (0 ≤ θ < 2π, t ∈ R2). (5)
We prove: there is no bounded measurable Steinhaus set.
Integrating (5) for t ∈ [0, 1]2 we obtain |E | = 1.
LHS of (5) is the Z2-periodization of 1RθE . Hence 1RθE (k) = 0,k ∈ Z2 \ {0}.1E (ξ) = 0, whenever ξ on a circle through a lattice point
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 33 / 36
![Page 182: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/182.jpg)
Application: the Steinhaus tiling problem
Question of Steinhaus: Is there E ⊆ R2 such that no matter howtranslated and rotated it always contains exactly one point withinteger coordinates?
Two versions: E is required to be measurable or not
Non-measurable version was answered in the affirmative by Jacksonand Mauldin a few years ago.
Measurable version remains open.
Equivalent form (Rθ is rotation by θ):∑k∈Z2
1RθE (t + k) = 1, (0 ≤ θ < 2π, t ∈ R2). (5)
We prove: there is no bounded measurable Steinhaus set.
Integrating (5) for t ∈ [0, 1]2 we obtain |E | = 1.
LHS of (5) is the Z2-periodization of 1RθE . Hence 1RθE (k) = 0,k ∈ Z2 \ {0}.1E (ξ) = 0, whenever ξ on a circle through a lattice point
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 33 / 36
![Page 183: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/183.jpg)
Application: the Steinhaus tiling problem
Question of Steinhaus: Is there E ⊆ R2 such that no matter howtranslated and rotated it always contains exactly one point withinteger coordinates?
Two versions: E is required to be measurable or not
Non-measurable version was answered in the affirmative by Jacksonand Mauldin a few years ago.
Measurable version remains open.
Equivalent form (Rθ is rotation by θ):∑k∈Z2
1RθE (t + k) = 1, (0 ≤ θ < 2π, t ∈ R2). (5)
We prove: there is no bounded measurable Steinhaus set.
Integrating (5) for t ∈ [0, 1]2 we obtain |E | = 1.
LHS of (5) is the Z2-periodization of 1RθE . Hence 1RθE (k) = 0,k ∈ Z2 \ {0}.1E (ξ) = 0, whenever ξ on a circle through a lattice point
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 33 / 36
![Page 184: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/184.jpg)
Application: the Steinhaus tiling problem
Question of Steinhaus: Is there E ⊆ R2 such that no matter howtranslated and rotated it always contains exactly one point withinteger coordinates?
Two versions: E is required to be measurable or not
Non-measurable version was answered in the affirmative by Jacksonand Mauldin a few years ago.
Measurable version remains open.
Equivalent form (Rθ is rotation by θ):∑k∈Z2
1RθE (t + k) = 1, (0 ≤ θ < 2π, t ∈ R2). (5)
We prove: there is no bounded measurable Steinhaus set.
Integrating (5) for t ∈ [0, 1]2 we obtain |E | = 1.
LHS of (5) is the Z2-periodization of 1RθE . Hence 1RθE (k) = 0,k ∈ Z2 \ {0}.
1E (ξ) = 0, whenever ξ on a circle through a lattice point
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 33 / 36
![Page 185: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/185.jpg)
Application: the Steinhaus tiling problem
Question of Steinhaus: Is there E ⊆ R2 such that no matter howtranslated and rotated it always contains exactly one point withinteger coordinates?
Two versions: E is required to be measurable or not
Non-measurable version was answered in the affirmative by Jacksonand Mauldin a few years ago.
Measurable version remains open.
Equivalent form (Rθ is rotation by θ):∑k∈Z2
1RθE (t + k) = 1, (0 ≤ θ < 2π, t ∈ R2). (5)
We prove: there is no bounded measurable Steinhaus set.
Integrating (5) for t ∈ [0, 1]2 we obtain |E | = 1.
LHS of (5) is the Z2-periodization of 1RθE . Hence 1RθE (k) = 0,k ∈ Z2 \ {0}.1E (ξ) = 0, whenever ξ on a circle through a lattice point
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 33 / 36
![Page 186: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/186.jpg)
The circles on which 1E must vanish
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 34 / 36
![Page 187: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/187.jpg)
Application: the Steinhaus tiling problem: conclusion
Consider the projection f of 1E on R.E bounded =⇒ f has compact support, say in [−B,B].
For ξ ∈ R we have f (ξ) = 1E (ξ, 0), hence
f(√
m2 + n2)
= 0, (m, n) ∈ Z2 \ {0}.
Landau: The number of integers up to x which are sums of twosquares is ∼ Cx/ log1/2 x .
Hence f has almost R2 zeros from 0 to R.
supp f ⊆ [−B,B] implies∣∣∣f (z)
∣∣∣ ≤ ‖f ‖1e2πB|z|, z ∈ C
But such a function can only have O(R) zeros from 0 to R.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 35 / 36
![Page 188: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/188.jpg)
Application: the Steinhaus tiling problem: conclusion
Consider the projection f of 1E on R.E bounded =⇒ f has compact support, say in [−B,B].
For ξ ∈ R we have f (ξ) = 1E (ξ, 0), hence
f(√
m2 + n2)
= 0, (m, n) ∈ Z2 \ {0}.
Landau: The number of integers up to x which are sums of twosquares is ∼ Cx/ log1/2 x .
Hence f has almost R2 zeros from 0 to R.
supp f ⊆ [−B,B] implies∣∣∣f (z)
∣∣∣ ≤ ‖f ‖1e2πB|z|, z ∈ C
But such a function can only have O(R) zeros from 0 to R.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 35 / 36
![Page 189: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/189.jpg)
Application: the Steinhaus tiling problem: conclusion
Consider the projection f of 1E on R.E bounded =⇒ f has compact support, say in [−B,B].
For ξ ∈ R we have f (ξ) = 1E (ξ, 0), hence
f(√
m2 + n2)
= 0, (m, n) ∈ Z2 \ {0}.
Landau: The number of integers up to x which are sums of twosquares is ∼ Cx/ log1/2 x .
Hence f has almost R2 zeros from 0 to R.
supp f ⊆ [−B,B] implies∣∣∣f (z)
∣∣∣ ≤ ‖f ‖1e2πB|z|, z ∈ C
But such a function can only have O(R) zeros from 0 to R.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 35 / 36
![Page 190: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/190.jpg)
Application: the Steinhaus tiling problem: conclusion
Consider the projection f of 1E on R.E bounded =⇒ f has compact support, say in [−B,B].
For ξ ∈ R we have f (ξ) = 1E (ξ, 0), hence
f(√
m2 + n2)
= 0, (m, n) ∈ Z2 \ {0}.
Landau: The number of integers up to x which are sums of twosquares is ∼ Cx/ log1/2 x .
Hence f has almost R2 zeros from 0 to R.
supp f ⊆ [−B,B] implies∣∣∣f (z)
∣∣∣ ≤ ‖f ‖1e2πB|z|, z ∈ C
But such a function can only have O(R) zeros from 0 to R.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 35 / 36
![Page 191: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/191.jpg)
Application: the Steinhaus tiling problem: conclusion
Consider the projection f of 1E on R.E bounded =⇒ f has compact support, say in [−B,B].
For ξ ∈ R we have f (ξ) = 1E (ξ, 0), hence
f(√
m2 + n2)
= 0, (m, n) ∈ Z2 \ {0}.
Landau: The number of integers up to x which are sums of twosquares is ∼ Cx/ log1/2 x .
Hence f has almost R2 zeros from 0 to R.
supp f ⊆ [−B,B] implies∣∣∣f (z)
∣∣∣ ≤ ‖f ‖1e2πB|z|, z ∈ C
But such a function can only have O(R) zeros from 0 to R.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 35 / 36
![Page 192: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/192.jpg)
Application: the Steinhaus tiling problem: conclusion
Consider the projection f of 1E on R.E bounded =⇒ f has compact support, say in [−B,B].
For ξ ∈ R we have f (ξ) = 1E (ξ, 0), hence
f(√
m2 + n2)
= 0, (m, n) ∈ Z2 \ {0}.
Landau: The number of integers up to x which are sums of twosquares is ∼ Cx/ log1/2 x .
Hence f has almost R2 zeros from 0 to R.
supp f ⊆ [−B,B] implies∣∣∣f (z)
∣∣∣ ≤ ‖f ‖1e2πB|z|, z ∈ C
But such a function can only have O(R) zeros from 0 to R.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 35 / 36
![Page 193: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/193.jpg)
Zeros of entire functions of exponential type
Jensen’s formula: F analytic in the disk {|z | ≤ R}, zk are the zerosof F in that disk. Then∑
k
logR
|zk |=
∫ 1
0log
∣∣∣F (Re2πiθ)∣∣∣ dθ.
It follows
#{k : |zk | ≤ R
/e}≤
∫ 1
0log
∣∣∣F (Re2πiθ)∣∣∣ dθ
Suppose |F (z)| ≤ AeB|z|. Then RHS above is ≤ BR + log A.
Such a function F can therefore have only O(R) zeros in the disk{|z | ≤ R}.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 36 / 36
![Page 194: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/194.jpg)
Zeros of entire functions of exponential type
Jensen’s formula: F analytic in the disk {|z | ≤ R}, zk are the zerosof F in that disk. Then∑
k
logR
|zk |=
∫ 1
0log
∣∣∣F (Re2πiθ)∣∣∣ dθ.
It follows
#{k : |zk | ≤ R
/e}≤
∫ 1
0log
∣∣∣F (Re2πiθ)∣∣∣ dθ
Suppose |F (z)| ≤ AeB|z|. Then RHS above is ≤ BR + log A.
Such a function F can therefore have only O(R) zeros in the disk{|z | ≤ R}.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 36 / 36
![Page 195: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/195.jpg)
Zeros of entire functions of exponential type
Jensen’s formula: F analytic in the disk {|z | ≤ R}, zk are the zerosof F in that disk. Then∑
k
logR
|zk |=
∫ 1
0log
∣∣∣F (Re2πiθ)∣∣∣ dθ.
It follows
#{k : |zk | ≤ R
/e}≤
∫ 1
0log
∣∣∣F (Re2πiθ)∣∣∣ dθ
Suppose |F (z)| ≤ AeB|z|. Then RHS above is ≤ BR + log A.
Such a function F can therefore have only O(R) zeros in the disk{|z | ≤ R}.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 36 / 36
![Page 196: The Fourier Transform and applications - Mihalis Kolountzakis](https://reader035.fdocuments.us/reader035/viewer/2022070222/613c5fbaf237e1331c513198/html5/thumbnails/196.jpg)
Zeros of entire functions of exponential type
Jensen’s formula: F analytic in the disk {|z | ≤ R}, zk are the zerosof F in that disk. Then∑
k
logR
|zk |=
∫ 1
0log
∣∣∣F (Re2πiθ)∣∣∣ dθ.
It follows
#{k : |zk | ≤ R
/e}≤
∫ 1
0log
∣∣∣F (Re2πiθ)∣∣∣ dθ
Suppose |F (z)| ≤ AeB|z|. Then RHS above is ≤ BR + log A.
Such a function F can therefore have only O(R) zeros in the disk{|z | ≤ R}.
Mihalis Kolountzakis (U. of Crete) FT and applications January 2006 36 / 36