Universal Taylor and trigonometric series in the sense of Menchoff
type: Item
instance of: scholarly article
title: Universal Taylor and trigonometric series in the sense of Menchoff
zbMATH Open document ID: 1028.42006
DOI: 10.1016/S0021-7824(00)00175-6
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2004-01-15T00:00:00Z
review text: A trigonometric series is said to be universal in the sense of Menchoff provided any periodic, measurable function \(f(t)\) exists as the almost everywhere limit of a suitable sequence of partial sums \(\lim_{j\to\infty} S_N(j)(t)\) of the given series. Menchoff proved the existence of universal series in 1945. This note provides a proof that such behavior is typical, that is, for almost every sequence \(\{c_n\}_{n\in\mathbb{N}}\), the series \(\sum_{n\in\mathbb{N}} c_n e^{ i nt}\) is universal. The proof relies on the ability to approximate continuous periodic functions uniformly outside of a subset of the torus having small measure from within a manifold of trigonometric polynomials whose sequences of coefficients are uniformly bounded both in \(\ell^2\) as well as \(\ell^\infty\). The authors go on to verify that a series can be universal even when a fairly strong condition is imposed on the decay of its coefficients. Precisely, suppose that \(\phi\) is a strictly increasing function on \(\mathbb{R}_+\) such that \(\phi(x)=O(x^2)\) as \(x\to 0\), or \(\phi\) is convex and \(\phi(2x)=O(\phi(x))\) as \(x\to 0\). Then there is a universal series \(\sum c_n e^{int}\) such that \(\sum_{n=0}^\infty \phi(|c_n|) <\infty\).
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 42A32
Mathematics Subject Classification ID: 30B30
zbMATH DE Number: 1546957
zbMATH Keywords: universal trigonometric series
zbMATH Keywords: Taylor series
zbMATH Keywords: polynomial approximation
author: Jean-Pierre Kahane
author: V. Nestoridis
reviewed by: Joseph Lakey
MaRDI profile type: Publication
full work available at URL: s0021-7824(00)00175-6
OpenAlex ID: W2002710602
Recommended article: Q4270264
Recommended article: Universal Taylor series and summability
Recommended article: Sur les séries de Taylor universelles
Recommended article: Universality and summability of trigonometric polynomials and trigonometric series
Recommended article: On various types of universal taylor series
Recommended article: On the existence of universal series by trigonometric system
Recommended article: A STRONG NOTION OF UNIVERSAL TAYLOR SERIES
Recommended article: Some remarks on universal functions and Taylor series
Recommended article: Pointwise universal trigonometric series
Recommended article: On the radial behavior of universal Taylor series
description: scientific article; zbMATH DE number 1546957