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The Department of Analysis of Eötvös Loránd University, PowerPoint Presentation

The Department of Analysis of Eötvös Loránd University,

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### Functions...

The Department of Analysis of Eötvös Loránd University,

PRESENTS

in cooperation with

Central European University,

and Limage Holding SA

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 difference functions f(x+h) - f(x) are continuous 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 grouph f = f(x+h) - f(x)

f difference functions f(x+h) - f(x) are continuous is measurable,

h fcontinuous

for everyhT

T

H(B,S)

fis continuous

B -measurable functions

S -continuous functions

Example on T :B difference functions f(x+h) - f(x) are continuous : L1 (T) S:L2(T)

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

- translation to general classes

(simple)

- done!

a difference functions f(x+h) - f(x) are continuous ie2πint

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 difference functions f(x+h) - f(x) are continuous

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:L difference functions f(x+h) - f(x) are continuous 1\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 difference functions f(x+h) - f(x) are continuous nonejectiveiff there is a > 0:

T

H

(A∆(A+h))=0

Nonejective sets

H(L1,L2)

Nonejective sets

H(L1,L2)

Nonejective sets:H difference functions f(x+h) - f(x) are continuous (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 difference functions f(x+h) - f(x) are continuous

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

Moreover:=

L difference functions f(x+h) - f(x) are continuous p

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 difference functions f(x+h) - f(x) are continuous (Lp,ACF)=

N

, 0<p<

, 0<p<

H(Lp,L )=F

H(Lip,Lip)

, 0<<<1,

classes coincide

Some other classes (T. Keleti):END

very complicated

H(B,C)

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