# Difference functions of periodic measurable functions

Fundamenta Mathematicae (1998)

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

## Access Full Article

top## Abstract

top## How to cite

topKeleti, 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

top- [1] M. Balcerzak, Z. Buczolich and M. Laczkovich, Lipschitz differences and Lipschitz functions, Colloq. Math. 72 (1997), 319-324. Zbl0877.26004
- [2] N. K. Bary, Trigonometric Series, Fizmatgiz, Moscow, 1961 (in Russian); English transl.: A Treatise on Trigonometric Series, Macmillan, New York, 1964.
- [3] N. G. de Bruijn, Functions whose differences belong to a given class, Nieuw Arch. Wisk. 23 (1951), 194-218. Zbl0042.28805
- [4] N. G. de Bruijn, A difference property for Riemann integrable functions and for some similar classes of functions, Indag. Math. 14 (1952), 145-151. Zbl0047.06202
- [5] L. Bukovský, N. N. Kholshchevnikova and M. Repický, Thin sets of harmonic analysis and infinite combinatorics, Real Anal. Exchange 20 (1994-1995), 454-509. Zbl0835.42001
- [6] B. Host, J-F. Méla and F. Parreau, Non singular transformations and spectral analysis of measures, Bull. Soc. Math. France 119 (1991), 33-90. Zbl0748.43001
- [7] S. Kahane, Antistable classes of thin sets in harmonic analysis, Illinois J. Math. 37 (1993), 186-223. Zbl0793.42003
- [8] T. Keleti, On the differences and sums of periodic measurable functions, Acta Math. Hungar. 75 (1997), 279-286. Zbl0929.26002
- [9] T. Keleti, Difference functions of periodic measurable functions, PhD thesis, Eötvös Loránd University, Budapest, 1996 (http://www.cs.elte.hu/phd_th/). Zbl0910.28003
- [10] T. Keleti, Periodic $Li{p}^{\alpha}$ functions with $Li{p}^{\beta}$ difference functions, Colloq. Math. 76 (1998), 99-103. Zbl0896.26005
- [11] T. Keleti, Periodic ${L}_{p}$ functions with ${L}_{q}$ difference functions, Real Anal. Exchange, to appear. Zbl0943.28002
- [12] M. Laczkovich, Functions with measurable differences, Acta Math. Acad. Sci. Hungar. 35 (1980), 217-235. Zbl0468.28006
- [13] M. Laczkovich, On the difference property of the class of pointwise discontinuous functions and some related classes, Canad. J. Math. 36 (1984), 756-768. Zbl0564.39001
- [14] M. Laczkovich and Sz. Révész, Periodic decompositions of continuous functions, Acta Math. Hungar. 54 (1989), 329-341.
- [15] M. Laczkovich and I. Z. Ruzsa, Measure of sumsets and ejective sets I, Real Anal. Exchange 22 (1996-97), 153-166.
- [16] W. Sierpiński, Sur les translations des ensembles linéaires, Fund. Math. 19 (1932), 22-28.
- [17] A. Zygmund, Trigonometric Series, Vols. I-II, Cambridge Univ. Press, 1959.