Sets of convergence to zero of lacunary trigonometric series
type: Item
instance of: scholarly article
title: Sets of convergence to zero of lacunary trigonometric series
zbMATH Open document ID: 0715.42014
DOI: 10.1007/BF01137634
published in: Mathematical Notes
publication date: 1989-01-01T00:00:00Z
review text: Let \(E\subset [0,2\pi)\) be a set such that any lacunary trigonometric series (in Hadamard's sense) \[ a_ 0/2+\sum^{\infty}_{k=1}a_ k \cos n_ kx+b_ k \sin n_ kx, \] where \(n_{k+1}/n_ k\geq q>1\) \((k=1,2,...)\), which converges to zero on E, is the identically zero series, i.e., \(a_ 0=a_ k=b_ k=0\) \((k=1,2,...)\). The set [0,2\(\pi\))\(\setminus E\) is called a set of uniqueness for lacunary series. We consider the question: Is a set of uniqueness necessarily of the cardinality of the continuum \({\mathfrak c}?\) It turns out that this problem is undecidable in Zermelo-Fraenkel set theory.
Mathematics Subject Classification ID: 42A63
Mathematics Subject Classification ID: 42A55
zbMATH DE Number: 4177833
zbMATH Keywords: lacunary trigonometric series
zbMATH Keywords: uniqueness for lacunary series
author: Natalia Kholshchevnikova
MaRDI profile type: Publication
cites work: Q3254471
cites work: Independence results concerning the number of nowhere dense sets necessary to cover the real line
cites work: Lectures in set theory with particular emphasis on the method of forcing
cites work: On the combinatorial principle P(c)
cites work: Q3874454
description: scientific article; zbMATH DE number 4177833