Brochure of Eötvös Loránd University - Faculty of Social Sciences
The Department of Analysis of Eötvös Loránd University,
description
Transcript of The Department of Analysis of Eötvös Loránd University,
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The Department of Analysis of Eötvös Loránd University, in cooperation with
Central European University,and Limage Holding SA
PRESENTPRESENTSS
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Functions...
Tamás Mátrai
...and their differencesDifferences...
Imre Ruzsa Miklós Laczkovich
HostBalcerzak
Parr
eau
Kahane Buczolich
Méla
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”If f is a measurable real function such that the difference functions f(x+h) - f(x) are continuous then f itself
is continuous.”
for every real h,for every real h,
How many h’s should we consider?
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T T : circle group
h f = f(x+h) - f(x)If B and S are two classes of real functions on TT with S B then
€
⊂
H(B,S)= H T T : there is an f B \ S
€
{
€
}€
⊂
€
∈
such that h f S for every h H
€
∈
€
∈
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Example on T T : B -measurable functionsS -continuous functions
f is measurable,
h f continuous
for every h T T
f is continuous
€
}€
∈
€
{ T T H(B,S)
€
∉
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Work schedule:
B: L1 (TT) S: L2(TT)
€
⊃
(simple)
• H(B,S) for special function classes;
• translation to general classes
• done!
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Upper bound for H(L1,L2):
H H(L1,L2)
€
∈f ~
€
∑ aie2πint
h f =
€
∑ai(e2πin(t+h)- e2πint) =
€
∑ ai e2πint(e2πinh -1)
€
∫ dµ(h)
€
∫ dµ(h)
€
∫ dµ(h)measure concentrated on H
€
∫ dµ(h)(e2πinh -1) > > 0?What if
||h f|| < 1L2
€
∈ H, h
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Weak Dirichlet sets:Borel set H is weak Dirichletweak Dirichlet if for every probability measure µconcentrated on H,
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⊂ T T
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∫ dµ(h) (e2πinh -1) = 0
€
liminfn →∞
weak Dirichlet sets
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⊂H(L1,L2)
weak Dirichlet sets
€
⊂H(L1,L2)
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Lower bound for H(L1,L2):
€
⊂ T T HWanted f
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∈L1\L2: h f
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∈ L2for everyh H
€
∈
Try characteristic functions!
€
⊂A T T , f =A
h f = f(x+h)-f(x) = =A(x+h)- A(x)= A∆(A+h)
What if (A)is big, while(A∆(A+h))is very small for every h H?
€
∈
symetric difference
Lebesgue measure
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Nonejective sets:
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⊂ T T H is nonejective iff there is a > 0:
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inf A⊂Tλ (A )=δ
suph∈H
(A∆(A+h))=0
Nonejective sets
€
⊂ H(L1,L2)
Nonejective sets
€
⊂ H(L1,L2)
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Some lemmas:
HostMéla
Parreau€
⊂ T T His anN-set iff it can becovered by a countable union of:weak Dirichlet sets
sets of absolute convregenceof not everywhere convergent
Fourier series
I. Ruzsa:Compact
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⊂ T T His weak Dirichlet iffit is nonejective.H(L1,L2) =N - setsH(L1,L2) =N - sets
T. Keleti: Every is a subset of an F subgroup of TT.
H(L1,L2) H
€
∈
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Moreover:
F€
∈={f L2:
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∫||f||L2= 1, TTf = 0}
M (H)={probability measures on H}
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inf f ∈F
suph∈H
||∆hf||L22
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sup μ∈M ( H )
infn≠0
€
∫TT|e2inh-1|2 dµ(h)
=
“A set is as ejective as far from being Weak Dirichlet.”
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Translation for other classes:Take powers: f
€
∈Lp f
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∈ L
€
hf
€
∈Lp hf
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∈L
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if >1
H(Lp,Lq) =N - sets
Only for 0
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≤
€
≤q
€
≤p 2:
H(Lp,Lq) =N - sets
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Some other classes (T. Keleti):H(Lp,ACF)=N , 0<p<
€
∞
€
∞
H(Lp,L )=F
€
∞ , 0<p<
€
∞
H(Lip,Lip) classes coincide, 0<<<1,
H(B,C)
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