Beurling moving averages and approximate homomorphisms

The theory of regular variation, in its Karamata and Bojanić–Karamata/de Haan forms, is long established and makes essential use of homomorphisms. Both forms are subsumed within the recent theory of Beurling regular variation, developed further here, especially certain moving averages occurring ther...

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Bibliographic Details
Published in:Indagationes mathematicae 2016-06, Vol.27 (3), p.601-633
Main Authors: Bingham, N.H., Ostaszewski, A.J.
Format: Article
Language:English
Subjects:
Beurling regular variation
Beurling’s functional equation
Category-measure duality
Circle group
Gołąb–Schinzel functional equation
Self-equivarying functions
Self-neglecting functions
Uniform convergence theorem
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Summary:The theory of regular variation, in its Karamata and Bojanić–Karamata/de Haan forms, is long established and makes essential use of homomorphisms. Both forms are subsumed within the recent theory of Beurling regular variation, developed further here, especially certain moving averages occurring there. Extensive use of group structures leads to an algebraicization not previously encountered here, and to the approximate homomorphisms of the title. Dichotomy results are obtained: things are either very nice or very nasty. Quantifier weakening is extended, and the degradation resulting from working with limsup and liminf, rather than assuming limits exist, is studied.
ISSN:0019-3577
1872-6100
DOI:10.1016/j.indag.2015.11.011