# Difference functions of periodic measurable functions

Fundamenta Mathematicae (1998)

• Volume: 157, Issue: 1, page 15-32
• ISSN: 0016-2736

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## Abstract

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We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions ${\Delta }_{h}f\left(x\right)=f\left(x+h\right)-f\left(x\right)$ are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, $ℌ\left(ℱ,G\right)=H\subset ℝ/ℤ:\left(\exists f\in ℱ\phantom{\rule{4pt}{0ex}}G\right)\left(\forall h\in H\right){\Delta }_{h}f\in G$, we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group $𝕋=ℝ/ℤ$ that are invariant for changes on null-sets (e.g. measurable functions, ${L}_{p}$, ${L}_{\infty }$, essentially continuous functions, functions with absolute convergent Fourier series (ACF*), essentially Lipschitz functions) and classes of continuous functions on $𝕋$ (e.g. continuous functions, continuous functions with absolute convergent Fourier series, Lipschitz functions). The classes ℌ(ℱ,G) are often related to some classes of thin sets in harmonic analysis (e.g. $ℌ\left({L}_{1},ACF*\right)$ is the class of N-sets). Some results concerning the difference property and the weak difference property of these classes of functions are also obtained.

## How to cite

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Keleti, Tamás. "Difference functions of periodic measurable functions." Fundamenta Mathematicae 157.1 (1998): 15-32. <http://eudml.org/doc/212274>.

@article{Keleti1998,
abstract = {We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions $Δ_h f(x)=f(x+h)-f(x)$ are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, $ℌ(ℱ,G) = \{H ⊂ ℝ/ℤ : (∃f ∈ ℱ \ G) (∀ h ∈ H) Δ_h f ∈ G\}$, we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group $\mathbb \{T\}=ℝ/ℤ$ that are invariant for changes on null-sets (e.g. measurable functions, $L_p$, $L_∞$, essentially continuous functions, functions with absolute convergent Fourier series (ACF*), essentially Lipschitz functions) and classes of continuous functions on $\mathbb \{T\}$ (e.g. continuous functions, continuous functions with absolute convergent Fourier series, Lipschitz functions). The classes ℌ(ℱ,G) are often related to some classes of thin sets in harmonic analysis (e.g. $ℌ(L_1,\{ACF\}*)$ is the class of N-sets). Some results concerning the difference property and the weak difference property of these classes of functions are also obtained.},
author = {Keleti, Tamás},
journal = {Fundamenta Mathematicae},
keywords = {periodic functions; difference functions; measurable functions; essentially continuous functions; essentially Lipschitz functions; functions with absolutely convergent Fourier series; ; ; -sets; pseudo-Dirichlet sets; difference property},
language = {eng},
number = {1},
pages = {15-32},
title = {Difference functions of periodic measurable functions},
url = {http://eudml.org/doc/212274},
volume = {157},
year = {1998},
}

TY - JOUR
AU - Keleti, Tamás
TI - Difference functions of periodic measurable functions
JO - Fundamenta Mathematicae
PY - 1998
VL - 157
IS - 1
SP - 15
EP - 32
AB - We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions $Δ_h f(x)=f(x+h)-f(x)$ are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, $ℌ(ℱ,G) = {H ⊂ ℝ/ℤ : (∃f ∈ ℱ \ G) (∀ h ∈ H) Δ_h f ∈ G}$, we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group $\mathbb {T}=ℝ/ℤ$ that are invariant for changes on null-sets (e.g. measurable functions, $L_p$, $L_∞$, essentially continuous functions, functions with absolute convergent Fourier series (ACF*), essentially Lipschitz functions) and classes of continuous functions on $\mathbb {T}$ (e.g. continuous functions, continuous functions with absolute convergent Fourier series, Lipschitz functions). The classes ℌ(ℱ,G) are often related to some classes of thin sets in harmonic analysis (e.g. $ℌ(L_1,{ACF}*)$ is the class of N-sets). Some results concerning the difference property and the weak difference property of these classes of functions are also obtained.
LA - eng
KW - periodic functions; difference functions; measurable functions; essentially continuous functions; essentially Lipschitz functions; functions with absolutely convergent Fourier series; ; ; -sets; pseudo-Dirichlet sets; difference property
UR - http://eudml.org/doc/212274
ER -

## References

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