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2017 | OriginalPaper | Buchkapitel

On the Uniform Theory of Lacunary Series

verfasst von: István Berkes

Erschienen in: Number Theory – Diophantine Problems, Uniform Distribution and Applications

Verlag: Springer International Publishing

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Abstract

The theory of lacunary series starts with Weierstrass’ famous example (1872) of a continuous, nondifferentiable function and now we have a wide and nearly complete theory of lacunary subsequences of classical orthogonal systems, as well as asymptotic results for thin subsequences of general function systems. However, many applications of lacunary series in harmonic analysis, orthogonal function theory, Banach space theory, etc. require uniform limit theorems for such series, i.e., theorems holding simultaneously for a class of lacunary series, and such results are much harder to prove than dealing with individual series. The purpose of this paper is to give a survey of uniformity theory of lacunary series and discuss new results in the field. In particular, we study the permutation-invariance of lacunary series and their connection with Diophantine equations, uniform limit theorems in Banach space theory, resonance phenomena for lacunary series, lacunary sequences with random gaps, and the metric discrepancy theory of lacunary sequences.

Vorheriges Kapitel On the Density of Coprime Tuples of the Form (n, ⌊ f 1(n)⌋, …, ⌊ f k (n)⌋), Where f 1, …, f k Are Functions from a Hardy Field

Nächstes Kapitel Diversity in Parametric Families of Number Fields

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Titel: On the Uniform Theory of Lacunary Series
verfasst von: István Berkes
Verlag: Springer International Publishing
Buch: Number Theory – Diophantine Problems, Uniform Distribution and Applications
Print ISBN: 978-3-319-55356-6

Electronic ISBN: 978-3-319-55357-3

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-55357-3_6

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