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Fundamenta Mathematicae

1998 | 157 | 1 | 15-32
Tytuł artykułu

Difference functions of periodic measurable functions

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
15-32
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-03-24
poprawiono
1998-01-08
Twórcy
autor
• Department of Analysis, Eötvös Loránd University, Múzeum krt. 6-8, 1088 Budapest, Hungary
Bibliografia
• [1] M. Balcerzak, Z. Buczolich and M. Laczkovich, Lipschitz differences and Lipschitz functions, Colloq. Math. 72 (1997), 319-324.
• [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.
• [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.
• [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.
• [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.
• [7] S. Kahane, Antistable classes of thin sets in harmonic analysis, Illinois J. Math. 37 (1993), 186-223.
• [8] T. Keleti, On the differences and sums of periodic measurable functions, Acta Math. Hungar. 75 (1997), 279-286.
• [9] T. Keleti, Difference functions of periodic measurable functions, PhD thesis, Eötvös Loránd University, Budapest, 1996 (http://www.cs.elte.hu/phd\_th/).
• [10] T. Keleti, Periodic $Lip^α$ functions with $Lip^β$ difference functions, Colloq. Math. 76 (1998), 99-103.
• [11] T. Keleti, Periodic $L_p$ functions with $L_q$ difference functions, Real Anal. Exchange, to appear.
• [12] M. Laczkovich, Functions with measurable differences, Acta Math. Acad. Sci. Hungar. 35 (1980), 217-235.
• [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.
• [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.
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Bibliografia
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