A problem of Orlicz in the Scottish Book
type: Item
instance of: scholarly article
title: A problem of Orlicz in the Scottish Book
zbMATH Open document ID: 0613.42004
author: Minoru Akita
author: Kazuo Goto
author: T. Kano
published in: Proceedings of the Japan Academy. Series A
publication date: 1986-01-01T00:00:00Z
review text: The problem mentioned in the title is to give an explicit example of a trigonometric series \(\sum (a_ n\cos (nx)+b_ n\sin (nx))\) that diverges everywhere, but for which \[ \sum^{\infty}_{n=1}(| a_ n|^{2+\epsilon}+| b_ n|^{2+\epsilon}) \] converges for every \(\epsilon >0\). As the authors note, the existence of such series is known, but no explicit construction had been given so far. In the paper under review the authors show that the series defined by \[ a_ n=\frac{\cos (2\pi \alpha n\quad \log n)}{\sqrt{n}},\quad b_ n=- \frac{\sin (2\pi \alpha n\quad \log n)}{\sqrt{n}} \] has the above properties for certain values of \(\alpha\) such as \(\alpha =1/\log 2\). The proof depends on the following exponential sum estimate: For N sufficiently large and real numbers \(\alpha >0\) and \(\beta\) the inequality \[ | \sum_{e^{-1/\alpha}N<n\leq N}\exp (2\pi i\alpha n\quad \log n+\beta n)| \geq C_{\alpha} \sqrt{N} \] holds with a positive constant \(C_{\alpha}\) depending only on \(\alpha\). The authors prove this estimate using van der Corput's method of transforming the exponential sum first into an integral and then evaluating this integral by the stationary phase method.
reviewed by: Adolf Hildebrand
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 42A20
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 3992102
zbMATH Keywords: estimates of exponential sum
zbMATH Keywords: van der Corput's method
zbMATH Keywords: stationary phase method
MaRDI profile type: Publication
full work available at URL: pjaa.62.267
OpenAlex ID: W2040533325
cites work: Q3944558
cites work: Q3715589
cites work: Q5338374
cites work: On Hardy-Littlewood Series
cites work: Asymptotic expansions of finite theta series
cites work: On a Polynomial of Byrnes
cites work: Q5803461
cites work: Error Bounds for Stationary Phase Approximations
DOI: 10.3792/PJAA.62.267
description: scientific article; zbMATH DE number 3992102