Limits of indeterminacy of trigonometric series
type: Item
instance of: scholarly article
title: Limits of indeterminacy of trigonometric series
zbMATH Open document ID: 0688.42003
DOI: 10.1007/BF01158029
published in: Mathematical Notes
publication date: 1988-01-01T00:00:00Z
review text: The author proves the following main theorem: for any trigonometric series: \(\sum^{\infty}_{n=0}c_ n \cos (nx+\theta_ n)\), let \(S_ n(x)=\sum^{n}_{m=0}c_ m \cos (mx+\theta_ m)\) and \(\bar S(x)= \limsup_{n\to \infty}S_ n(x)=F(x)\), \(\bar S(x)= \limsup_{n\to \infty}S_ n(x)=g(x)\), then \(mes\{x\in [-\pi,\pi]:\) \(-\infty <\underline S(x)\leq S(x)=+\infty \}=0.\) In particular \(mes\{x\in [-\pi,\pi]:\) \(\lim S(x)=\infty \}=0.\) This theorem is closely related to the work of \textit{D. E. Menchoff} [C. R. Acad. Sci. URSS, Nov. Ser. 26, 214-216 (1940; Zbl 0023.02901)]. By the end of this paper the author poses four interesting problems related to the above theorem.
Mathematics Subject Classification ID: 42A20
zbMATH DE Number: 4127837
author: Sergei Konyagin
MaRDI profile type: Publication
cites work: Q5784202
cites work: On Boundedness of Trigonometric Series
cites work: Q5734630
cites work: Martingale Theory and Pointwise Convergence of Certain Orthogonal Series
cites work: Q5574798
cites work: Sur les coefficients des séries de Fourier dont les sommes partielles sont positives sur un ensemble
full work available at URL: https://doi.org/10.1007/bf01158029
OpenAlex ID: W2029950158
description: scientific article; zbMATH DE number 4127837