The Department of Analysis of Eötvös Loránd University,

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The Department of Analysis of Eötvös Loránd University,

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The Department of Analysis of Eötvös Loránd University,

PRESENTS

in cooperation with

Central European University,

and Limage Holding SA

Balcerzak

Functions...

Méla

Differences...

Host

...and their differences

Tamás Mátrai

Kahane

Keleti

Buczolich

Parreau

Imre Ruzsa

Miklós Laczkovich

for every real h,

for every real h,

then f itself is continuous.”

How many h’s

should we consider?

If B and S are two classes of real functions onTwith S B then

H(B,S)= H T : there is an fB \S

such that h f S for every h H

h f = f(x+h) - f(x)

fis measurable,

h fcontinuous

for everyhT

T

H(B,S)

fis continuous

B -measurable functions

S -continuous functions

B: L1 (T)S:L2(T)

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

- translation to general classes

(simple)

- done!

aie2πint

f ~

H H(L1,L2)

H

||h f|| < 1

, h

L2

ai(e2πin(t+h)- e2πint) =

h f =

dµ(h)

dµ(h)

ai e2πint(e2πinh -1)

dµ(h)

measure

concentrated on H

(e2πinh -1)

> > 0?

dµ(h)

What if

T

Borel set H is weak Dirichlet

if for every probability measure µ

concentrated on H,

(e2πinh -1)

=0

dµ(h)

H(L1,L2)

H(L1,L2)

weak Dirichlet sets

weak Dirichlet sets

L1\L2:

Wanted f

L2

h f

for every

T

H

symetric

difference

h H

A

T

A

, f =

h f = f(x+h)-f(x) =

=A(x+h)-A(x)=A∆(A+h)

(A∆(A+h))

is very small for every h H?

What if

(A)

is big, while

Try characteristic functions!

Lebesgue measure

is nonejectiveiff there is a > 0:

T

H

(A∆(A+h))=0

Nonejective sets

H(L1,L2)

Nonejective sets

H(L1,L2)

H(L1,L2)

Every is a subset of an

F subgroup ofT.

H

T. Keleti:

sets of absolute convregence

of not everywhere convergent

Fourier series

is anN-set iff it can be

T

T

Host

Méla

Parreau

H

H

:

covered by a countable union of

weak Dirichlet sets

Compact

is weak Dirichlet iff

I. Ruzsa:

it is nonejective.

H(L1,L2) =N - sets

H(L1,L2) =N - sets

“A set is as ejective as far

from being Weak Dirichlet.”

F

= 1,

||f||L

={f L2:

}

f = 0

2

T

M (H)

=

{probability measures on H}

|e2inh-1|2 dµ(h)

T

2

||∆hf||L

2

=

Lp

Lp

f

f

L

L

hf

hf

if >1

p

Only for 0

q

2:

Take powers:

H(Lp,Lq) =N - sets

H(Lp,Lq) =N - sets

H(Lp,ACF)=

N

, 0<p<

, 0<p<

H(Lp,L )=F

H(Lip,Lip)

, 0<<<1,

classes coincide

END

very complicated

H(B,C)