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# The Department of Analysis of Eötvös Loránd University, PowerPoint PPT Presentation

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 M átrai. Kahane. Keleti. Buczolich. Parreau. Imre Ruzsa.

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

### ”If f is a measurable real function such that the difference functions f(x+h) - f(x) are continuous

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

### T :circle group

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)

### Work schedule:

• 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

### Upper bound for H(L1,L2):

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

### 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

### Lower bound for H(L1,L2):

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)

### Nonejective sets:

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.

### Some lemmas:

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}

|e2inh-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:

### Translation for other classes:

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)