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2017 | OriginalPaper | Chapter

On the Uniform Theory of Lacunary Series

Author : István Berkes

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

Publisher: 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.

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previous chapter 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

next chapter Diversity in Parametric Families of Number Fields

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Title: On the Uniform Theory of Lacunary Series
Author: István Berkes
Publisher: Springer International Publishing
Book: Number Theory – Diophantine Problems, Uniform Distribution and Applications
Print ISBN: 978-3-319-55356-6

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

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

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