Stability of group homomorphisms in the compact-open topology

Zlatoš, Pavol

doi:10.4115/jla.2010.2.3

Cited by 6 publications

(5 citation statements)

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“…Skof [110] studied the stability of the Cauchy equation on an interval, Kominek [69] investigated the equation on an N -dimensional cube in the space R N , and Sikorska [100,101] dealt with such stability postulated for orthogonal vectors in a ball centered at the origin. A somewhat more abstract approach to the conditional stability is presented in [116,117].…”

mentioning

confidence: 99%

Recent developments of the conditional stability of the homomorphism equation

Brzdęk¹,

Fechner²,

Moslehian³

et al. 2015

Banach J. Math. Anal.

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The issue of Ulam's type stability of an equation is understood in the following way: when a mapping which satisfies the equation approximately (in some sense), it is "close" to a solution of it. In this expository paper, we present a survey and a discussion of selected recent results concerning such stability of the equations of homomorphisms, focussing especially on some conditional versions of them. Some historyThis is an expository paper, but because of the demands of the journal, we had to restrict very significantly the number of references. We apologize to the authors of all the papers that are connected with the subjects considered here, but had to be omitted.The property of additivity of a mapping is very important (not only in mathematics) and can be described by the following Cauchy functional equationwhere f is a mapping between semigroups endowed with binary operations denoted by + (we abuse the notation and we use the same symbol for operations in two different structures). Every mapping satisfying equation (1.1) is said to be

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mentioning

confidence: 99%

Recent developments of the conditional stability of the homomorphism equation

Brzdęk¹,

Fechner²,

Moslehian³

et al. 2015

Banach J. Math. Anal.

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“…Analogous remarks could be made on behalf of the indicated generalizations of Theorems 2.4.1 and 2.4.2 and their relation to some standardly formulated stability results for characters of LCA groups with respect to the compact-open topology from Mačaj and Zlatoš [33] and Zlatoš [48]. However, the big amount of technicalities needed for their formulation and comparison would take us too far from the main lines and goals of the present paper.…”

Section: It Suffices To Show That For Each Symmetric Internal Setmentioning

confidence: 84%

Gordon's conjectures 1 and 2: Pontryagin-van Kampen duality in the hyperfinite setting

Zlatoš

2021

J. Log. Anal.

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“…In other words, such a generalization of 2.4.3 adds a considerable uniformness to the above consequence of the Pontryagin-van Kampen Duality Theorem. Analogous remarks could be made on behalf of the indicated generalizations of Theorems 2.4.1 and 2.4.2 and their relation to some standardly formulated stability results for characters of LCA groups with respect to the compact-open topology from [MZ] and [Zl2]. However, the big amount of technicalities needed for their formulation and comparison would take us too far from the main lines and goals of the present paper.…”

Section: Some (Mainly) Standard Equivalents: Hrushovski Style Theoremsmentioning

confidence: 85%

Gordon's Conjectures: Pontryagin-van Kampen duality and the Fourier transform in hyperfinite setting

Zlatos

2014

Preprint

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Using the ideas of E. I. Gordon we present and farther advance an approach, based on nonstandard analysis, to simultaneous approximations of locally compact abelian groups and their duals by (hyper)finite abelian groups, as well as to approximations of various types of Fourier transforms on them by the discrete Fourier transform. Combining some methods of nonstandard analysis and additive combinatorics we prove the three Gordon's Conjectures which were open since 1991 and are crucial both in the formulations and proofs of the LCA groups and Fourier transform approximation theorems. Contents 0. Introduction 1. Nonstandard analysis 1.1. General setting 1.2. Bounded monadic spaces 1.3. Functional spaces 1: Continuous functions 1.4. Functional spaces 2: Loeb measures and Lebesgue spaces 1.5. Bounded monadic groups 2. Pontryagin-van Kampen duality in hyperfinite setting 2.1. The dual triplet 2.2. Fourier transforms, Bohr sets and spectral sets in finite abelian groups 2.3. The dual triplet continued: Normal multipliers and proofs of Gordon's Conjectures 1 and 2 2.4. Some (mainly) standard equivalents: Hrushovski style theorems 2.5. Simultaneous approximation of an LCA group and its dual 3. The Fourier transform in hyperfinite dimensional setting 3.1. A characterization of liftings 3.2. The Smoothness-and-Decay Principle 3.3. Hyperfinite dimensional approximations of the Fourier transforms: Generalized Gordon's Conjecture 3 3.4. Standard consequences: Simultaneous approximation of a function and its Fourier transform References

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Stability of group homomorphisms in the compact-open topology

Cited by 6 publications

References 12 publications

Recent developments of the conditional stability of the homomorphism equation

Recent developments of the conditional stability of the homomorphism equation

Gordon's conjectures 1 and 2: Pontryagin-van Kampen duality in the hyperfinite setting

Gordon's Conjectures: Pontryagin-van Kampen duality and the Fourier transform in hyperfinite setting

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