The boundedness of the L1-norm of Walsh-Fejér...

The boundedness of the L1-norm of Walsh-Fejérkernels

Rodolfo Toledo

University of Nyı́regyháza

August 26, 2016

6th Workshop on Fourier Analysis and Related Fields

Pécs, Hungary

R. Toledo (University of Nyı́regyháza) The boundedness of the L1-norm of ... 6th Workshop on Fourier... 1 / 24

Trigonometric system

The L1-norm of kernels corresponding to operators related toorthonormal systems plays an important role in the convergence oforthogonal series.

Dirichlet kernels:

Dn(x) =1

2π+

1π

n∑k=1

cos k(x) =1π·

sin(n + 12)(x)2 sin 12(x)

Lebesgue constants:

Ln =∫ 2π

0|Dn(x)|dx =

2π

∫ π2

0

∣∣∣∣sin(2n + 1)tsin t∣∣∣∣ dt



Fejér (1910)

Ln =4π2

log n + a0 + εn (εn → 0).

Thus Ln = O(log n).

Gábor Szegő (1921)

Ln =16π2

∞∑k=1

14k2 − 1

(1 +

13

+15

+ · · ·+ 12k(2n + 1)− 1

)



Fejér kernels:

Knf (x) :=D0f (x) + D1f (x) + · · ·+ Dnf (x)

n + 1

=1

2(n + 1)π

(sin (n+1)(x)2

sin x2

)2≥ 0,

However ∫ 2π0|Kn(t , x)|dt = 1


The Walsh-Paley system

The binary expansion n = (n0,n1, . . . ) of the number n ∈ N:

n =∞∑

k=0

nk2k , (nk = 0 or nk = 1)

The binary expansion x = (x0, x1, . . . ) of the number x ∈ [0,1[:

x =∞∑

k=0

xk2k+1

, (xk = 0 or xk = 1)

The Rademacher system:

rk (x) := (−1)xk ,

where k ∈ N and x ∈ [0,1[.Figure: Rademacher function r3

The Rademacher system is orthonormal, but not complete.R. Toledo (University of Nyı́regyháza) The boundedness of the L1-norm of ... 6th Workshop on Fourier... 5 / 24


Walsh functions:The finite product of Rademacher functions.

The Walsh-Paley system:

ωn(x) :=∞∏

k=0

rnkk (x) = (−1)∑∞

k=0 xk nk (x ∈ [0,1[,n ∈ N).

The Walsh-Paley system is an orthonormal and complete systemdefined on the interval [0,1[.



Walsh-Dirichlet kernels:

Dn(x) :=n−1∑k=0

ωk (x) (x ∈ [0,1[)

Dyadic intervals:

Ik (i) :=[

i2k,i + 12k

[(i = 0, . . . ,2k − 1), Ik := Ik (0)

Paley’s lemma:

D2k (x) =

{2k , x ∈ Ik ,0, x ∈ [0,1[\Ik .

Figure: Dirichlet kernel D8


Walsh-Lebesgue constants

Walsh-Lebesgue constants:

Ln :=∫ 1

0|Dn(x)|dx (n ∈ N).

Figure: Walsh-Lebesgue constants


Walsh-Lebesgue constants

Fine (1949)Iteration for Walsh-Lebesgue constants: If n = 2k + m where0 ≤ m < 2k , then

Ln = 1 + Lm −m2k.

Properties (Fine)Ln = O(log n)L2n = Ln

L2n+1 =1 + Ln + Ln+1

21n∑n

k=1 Lk ≥ C log n


The L1-norm of Walsh-Fejér kernels

Walsh-Fejér kernels: If x ∈ [0,1[

Kn(x) :=1n

n∑k=1

Dk (x), ‖Kn‖1 :=∫ 1

0|Kn(x)|dx

Figure: The L1-norm of Walsh-Fejér kernels

The property ‖Kn‖1 ≤ 2 holds for all n ∈ P.R. Toledo (University of Nyı́regyháza) The boundedness of the L1-norm of ... 6th Workshop on Fourier... 10 / 24

The L1-norm of Walsh-Fejér kernels

Yano (1951)

K2k (x) =

2k+1

2 , x ∈ Ik ,2j−1, x ∈ Ik (2k−j−1), j = 0,1, . . . , k − 10, elsewhere in [0,1[.

Note that if x ∈ Ik (2k−j−1), then x = (0, . . . ,0︸︷︷︸j

,1,0, . . . ,0︸︷︷︸k−j−1

, xk , xk+1, . . . )

Figure: K32 Walsh-Fejér kernel K32


Iteration for the L1-norm of Walsh-Fejér kernels

Iteration for the L1-norm of Walsh-Fejér kernels: It was unknown untilnow.

The idea: Let n < 2k+1. Let us look at the values of Dn(x) when K2k (x)is positive.

LemmaLet k ,n ∈ N such that n < 2k+1.

If x ∈ Ik+1 then Dn(x) = n.If x ∈ Ik \ Ik+1 then

Dn(x) =

{n, n < 2k ,2k+1 − n, 2k ≤ n < 2k+1.



Lemma (continue)

Suppose x ∈ Ik (2k−j−1) for some j = 0,1, . . . , k − 1 and let p andn′ be non-negative integers such that n = p2j+1 + n′ holds, where0 ≤ n′ < 2j+1. If n < 2k or also, if n ≥ 2k but x is in the first half ofthe interval Ik (2k−j−1) then

Dn(x) =

{n′, n′ < 2j ,2j+1 − n′, 2j ≤ n′ < 2j+1.

Moreover, if n ≥ 2k and x is in the second half of the intervalIk (2k−j−1) then

Dn(x) =

{−n′, n′ < 2j ,n′ − 2j+1, 2j ≤ n′ < 2j+1.



Corollary

If n < 2k+1, then the value of Kn(x) is nonnegative if the value ofK2k (x) is positive.

LemmaLet k ,n ∈ N such that n < 2k+1.

If x ∈ Ik+1 then

nKn(x) =n(n + 1)

2.

If x ∈ Ik \ Ik+1 then

nKn(x) =

{n(n+1)

2 , n < 2k ,

n(n+1)2 − (n − 2

k )(n − 2k + 1), 2k ≤ n < 2k+1.



Lemma (continue)

Suppose x ∈ Ik (2k−j−1) for some j = 0,1, . . . , k − 1 and let p andn′ be non-negative integers such that n = p2j+1 + n′ holds, where0 ≤ n′ < 2j+1. If n < 2k or also, if n ≥ 2k but x is in the first half ofthe interval Ik (2k−j−1) then

nKn(x) =

{p4j + n

′(n′+1)2 , n

′ < 2j ,p4j + n

′(n′+1)2 − (n

′ − 2j)(n′ − 2j + 1), 2j ≤ n′ < 2j+1.

Moreover, if n ≥ 2k and x is in the second half of the intervalIk (2k−j−1) then

nKn(x) =

{2k+j − p4j − n

′(n′+1)2 , n

′ < 2j ,2k+j − p4j − n

′(n′+1)2 + (n

′ − 2j)(n′ − 2j + 1), 2j ≤ n′ < 2j+1.



New notation:

n′j :=j∑

i=0

ni2i = n mod 2j+1,

where j is a nonnegative integer.

Theorem

Let k ∈ N and n be a positive integer such that 2k ≤ n < 2k+1. Then,the following iteration is valid:

n‖Kn‖1 =n + n′k−1‖Kn′k−1‖1 −Γn

2k+1,

where

Γn := 2n′k−1(n′k−1 + 1) +

k−1∑j=1

(nj2j(2j − 1) + (1− 2nj)n′j−1(n

′j−1 + 1)

).



New notation: Let n = 2k0 + 2k1 + · · ·+ 2ks , where k0 < k1 < · · · < ks.Denote

n′ki =i∑

r=0

2kr

where i = 0,1, . . . , s.

Theorem

Suppose that n = 2k0 + 2k1 + · · ·+ 2ks , where k0 < k1 < · · · < ks arenonnegative integers. Then

n‖Kn‖1 =n + n′ks−1‖Kn′ks−1‖1 −

Γn2ks+1

,

where

Γn = n′ks−1(3n′ks−1 + 2) +

s−1∑i=0

(4ki + (ki+1 − ki − 2)n′ki (n

′ki + 1)

).


Properties of the L1-norm of Walsh-Fejér kernels

Let n = 2k0 + 2k1 + · · ·+ 2ks and denote

γn := 2n′ks−1 +s−1∑i=0

(ki+1 − ki − 2)n′ki (n ∈ P)

Lemma

γn ≥ 0 and γn = 0 if and only if n is a power of 2.

Theorem

‖K2n‖1 − ‖Kn‖1 =1

4n

(γn′k02k0

+γn′k12k1

+ · · ·+γn′ks2ks

)≥ 0

and the equality holds if and only if n is a power of 2.



Theorem

Let k ∈ N. If n,m ≥ 2k and n + m = 3 · 2k − 1, then

n‖Kn‖1 −m‖Km‖1 = n −m.



For a fixed positive integer k denote by n∗(k) the index less than 2k+1

such that‖Kn∗(k)‖1 = max{‖Kn‖1 : 1 ≤ n < 2k+1}.

Theorem

Let k ∈ P. Then1 If k is even then n∗(k) = 1 + 22 + 24 + · · ·+ 2k and

‖Kn∗(k)‖1 =1715− s + 1

4s+1 − 1+

15 · 4s

,

where s = k2 .

2 If k is odd then n∗(k) = 21 + 23 + · · ·+ 2k and

‖Kn∗(k)‖1 =1715− 1

2s + 1

4s+1 − 1+

130 · 4s

,

where s = k−12 .



Theorem

sup{‖Kn‖1 : n ∈ P} =1715


Publication

https://doi.org/10.1016/j.jmaa.2017.07.075


Thank you for your attention.


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