Invariant decomposition of functions with respect to commuting invertible transformations

Farkas, Bálint; Harangi, Viktor; Keleti, Tamás; Révész, Szilárd György

doi:10.1090/s0002-9939-07-09267-2

Cited by 7 publications

(9 citation statements)

References 9 publications

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“…Obvious. The fact that the identity in R can be expressed as the sum of two periodic functions was pointed out by Mortola and Peirone in [16], where (AC) is used to obtain periodic functions from R to R (see also [5]). …”

Section: Additive Functions First Examplesmentioning

confidence: 99%

Discontinuous additive functions: Regular behavior vs. pathological features

Bernardi

2015

Expositiones Mathematicae

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This paper deals with properties of discontinuous additive functions (a function f is said to be additive if f(x + y) = f(x) + f(y) for all x and y). We refer to two different frameworks, namely Q[sqrt 2] (where the axiom of choice is not needed) and the whole set R. We construct an additive function which is both periodic and quasiperiodic (in the sense of definition 1), as well as two periodic functions whose sum is the identity function (see also [9]). We establish several properties and characterizations of additive functions: among these, the fact that the graph of an additive function is homogeneous, and that every straight line passing through a point of the graph divides it into two congruent parts. We introduce two metaphors to describe such properties informally. Several other aspects are discussed, such as arcwise connectedness and Lebesgue measurability. Lastly, we examine the consequences of changing the topology on R

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Section: Additive Functions First Examplesmentioning

confidence: 99%

Discontinuous additive functions: Regular behavior vs. pathological features

Bernardi

2015

Expositiones Mathematicae

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“…The proof of the following is the same as for Theorem 4.9 with the new aspect that taking averages in Γ requires some additional care. [9]). Let A, Γ be torsion free Abelian groups and a 1 , .…”

Section: Decompositions On Groupsmentioning

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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“…2 Added in proof : Since in [4] a positive answer was given to Question 0.2, it turned out that the answer to Question 3.1 is also affirmative.…”

Section: Proofmentioning

confidence: 99%

“…It started in the seventies with some unpublished work of I.Z. Ruzsa and continued among others in [1,2,[4][5][6][8][9][10][11][12][13] and [14]. If f = f 1 + · · · + f n is an (a 1 , .…”

Section: Introductionmentioning

confidence: 99%

Periodic decomposition of measurable integer valued functions

Keleti

2008

Journal of Mathematical Analysis and Applications

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We study those functions that can be written as a sum of (almost everywhere) integer valued periodic measurable functions with given periods. We show that being (almost everywhere) integer valued measurable function and having a real valued periodic decomposition with the given periods is not enough. We characterize those periods for which this condition is enough. We also get that the class of bounded measurable (almost everywhere) integer valued functions does not have the so-called decomposition property. We characterize those periods a 1 , . . . , a k for which an almost everywhere integer valued bounded measurable function f has an almost everywhere integer valued bounded measurable (a 1 , . . . , a k )-periodic decomposition if and only if a 1 · · · a k f = 0, where a f (x) = f (x + a) − f (x).

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Invariant decomposition of functions with respect to commuting invertible transformations

Cited by 7 publications

References 9 publications

Discontinuous additive functions: Regular behavior vs. pathological features

Discontinuous additive functions: Regular behavior vs. pathological features

The Periodic Decomposition Problem

Periodic decomposition of measurable integer valued functions

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