The Department of Analysis of Eötvös Loránd University, in cooperation with

Central European University,and Limage Holding SA

PRESENTPRESENTSS

Functions...

Tamás Mátrai

...and their differencesDifferences...

Imre Ruzsa Miklós Laczkovich

HostBalcerzak

Parr

eau

Kahane Buczolich

Méla

”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?

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

€

∈

€

∈

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)

€

∉

Work schedule:

B: L1 (TT) S: L2(TT)

€

⊃

(simple)

• H(B,S) for special function classes;

• translation to general classes

• done!

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

Weak Dirichlet sets:Borel set H is weak Dirichletweak Dirichlet if for every probability measure µconcentrated on H,

€

⊂ T T

€

∫ dµ(h) (e2πinh -1) = 0

€

liminfn →∞

weak Dirichlet sets

€

⊂H(L1,L2)

weak Dirichlet sets

€

⊂H(L1,L2)

Lower bound for H(L1,L2):

€

⊂ T T HWanted f

€

∈L1\L2: h f

€

∈ 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

Nonejective sets:

€

⊂ T T H is nonejective iff there is a > 0:

€

inf A⊂Tλ (A )=δ

suph∈H

(A∆(A+h))=0

Nonejective sets

€

⊂ H(L1,L2)

Nonejective sets

€

⊂ H(L1,L2)

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

€

⊂ 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

€

∈

Moreover:

F€

∈={f L2:

€

∫||f||L2= 1, TTf = 0}

M (H)={probability measures on H}

€

inf f ∈F

suph∈H

||∆hf||L22

€

sup μ∈M ( H )

infn≠0

€

∫TT|e2inh-1|2 dµ(h)

=

“A set is as ejective as far from being Weak Dirichlet.”

Translation for other classes:Take powers: f

€

∈Lp f

€

∈ L

€

hf

€

∈Lp hf

€

∈L

€

if >1

H(Lp,Lq) =N - sets

Only for 0

€

≤

€

≤q

€

≤p 2:

H(Lp,Lq) =N - sets

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)

ENDEND