Periodic decomposition of measurable integer valued functions

Abstract. Consider a 1 , . . . , a n ∈ R arbitrary elements. We characterize those functions f : R → R that decompose into the sum of a j -periodic functions, i.e., f = f 1 +· · ·+f n with ∆ a j f (x) := f (x+a j )−f (x) = 0. We show that f has such a decomposition if and only if for all partitions B 1 ∪B 2 ∪· · ·∪B N = {a 1 , . . . , a n } with B j consisting of commensurable elements with least common multiplesActually, we prove a more general result for periodic decompositions of functions f : A → R defined on an Abelian group A; in fact, we even consider invariant decompositions of functions f : A → R with respect to commuting, invertible self-mappings of some abstract set A.We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.

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“…A combination of our results with [8] yields a characterization (Corollary 4.3) of those periods a 1 , . .…”

Section: Now Consider Pairwise Commuting Transformationsmentioning

confidence: 64%

“…Finally, for measurable decompositions we can draw the following consequence of our results. Proposition 3.3 in [8] …”

mentioning

confidence: 99%

Invariant decomposition of functions with respect to commuting invertible transformations

Farkas

Harangi

Keleti

et al. 2007

Proc. Amer. Math. Soc.

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“…For classes of measurable real functions we have, e.g., the following. For more information on measurable decompositions see also [23,24,25]. Next we turn to integer-valued decompositions on Abelian groups.…”

Section: Further Resultsmentioning

confidence: 99%

“…T. Keleti[26]). None of the following classes F have the decomposition property:a) F = {f : f : R → Z, f ∈ L ∞ (R)}, b) F = {f : f : R → Z is bounded measurable}, c) F = {f : f : R → R is a.e.…”

mentioning

confidence: 99%

The Periodic Decomposition Problem

Farkas

Révész

2014

Springer Proceedings in Mathematics &Amp; Statistics

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Dedicated to Imre Z. Ruzsa on the occasion of his 60 th birthday.Abstract If a function f : R → R can be represented as the sum of n periodic functions as f = f 1 + · · · + f n with f (x + α j ) = f (x) (j = 1, . . . , n), then it also satisfies a corresponding n-order difference equation ∆ α1 . . . ∆ αn f = 0. The periodic decomposition problem asks for the converse implication, which may hold or fail depending on the context (on the system of periods, on the function class in which the problem is considered, etc.). The problem has natural extensions and ramifications in various directions, and is related to several other problems in real analysis, Fourier and functional analysis. We give a survey about the available methods and results, and present a number of intriguing open problems.

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“…Several natural questions remain open. Measurable integer valued periodic decompositions of functions are studied in a subsequent paper [9] of the second author.…”

Section: Introductionmentioning

confidence: 99%

Periodic decomposition of integer valued functions

et al. 2008

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We study those functions that can be written as a nite sum of periodic integer valued functions. On Z we give three dierent characterizations of these functions. For this we prove that the existence of a real valued periodic decomposition of a Z → Z function implies the existence of an integer valued periodic decomposition with the same periods. This result depends on the representation of the greatest common divisor of certain polynomials with integer coecients as a linear combination of the given polynomials where the coecients also belong to Z[x]. We give an example of an R → {0, 1} function that has a bounded real valued periodic decomposition but does not have a bounded integer valued periodic decomposition with the same periods. It follows that the class of bounded Z → Z functions has the decomposition property as opposed to the class of bounded R → Z functions. If the periods are pairwise commensurable or not prescribed, then we get more general results.

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Periodic decomposition of measurable integer valued functions

Cited by 5 publications

References 8 publications

Invariant decomposition of functions with respect to commuting invertible transformations

Invariant decomposition of functions with respect to commuting invertible transformations

The Periodic Decomposition Problem

Periodic decomposition of integer valued functions

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