On the uniform theory of lacunary series

István Berkes*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

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.

Original languageEnglish
Title of host publicationNumber Theory - Diophantine Problems, Uniform Distribution and Applications
Subtitle of host publicationFestschrift in Honour of Robert F. Tichy's 60th Birthday
PublisherSpringer International Publishing AG
Pages137-167
Number of pages31
ISBN (Electronic)9783319553573
ISBN (Print)9783319553566
DOIs
Publication statusPublished - 1 Jun 2017

ASJC Scopus subject areas

  • Mathematics(all)

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