On the Uniform Theory of Lacunary Series

István Berkes ()
Additional contact information
István Berkes: Institute of Statistics, Graz University of Technology

A chapter in Number Theory – Diophantine Problems, Uniform Distribution and Applications, 2017, pp 137-167 from Springer

Abstract: 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.

Keywords: Limit Theorem; Central Limit Theorem; Limit Distribution; Diophantine Equation; Iterate Logarithm (search for similar items in EconPapers)
Date: 2017
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-55357-3_6

Ordering information: This item can be ordered from
http://www.springer.com/9783319553573

DOI: 10.1007/978-3-319-55357-3_6

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().