Absolute summability of Fourier series and orthogonal series
type: Item
instance of: scholarly article
title: Absolute summability of Fourier series and orthogonal series
zbMATH Open document ID: 0539.42013
author: Yasuo Okuyama
published in: Lecture Notes in Mathematics
publication date: 1984-01-01T00:00:00Z
review text: The successive lecture notes in mathematics deal with absolute Nörlund summability \(| N,p_ n|\) and absolute Riesz summability \(| R,P_ n,1|\) for non-absolute convergent Fourier series and orthogonal series. In this study, starting with the classical results of S. Bernstein, A. Zygmund, R. Salem, O. Szász and S. B. Stechkin, the author presents his own resuls (up to 1981), as well as those obtained by Ch. Watari, S. N. Lal, T. Tsuchikura, T. Kano, G. D. Dikshit, M. Izumi, S. Izumi and other mathematicians. The comprehensive references contain 96 positions, mainly sources. This book consists of 8 chapters the titles of which inform us about the essence, namely: Chapter 1, Absolute convergence of orthogonal series (introduction, comparison theorem, weight absolute convergence, convergence of \(\sum | c_ n|^{\beta})\); Chapter 2, Absolute Nörlund summability almost everywhere of Fourier series (introduction, analogous theorem, proof of theorem, approximation property and absolute Nörlund summability); Chapter 3, Absolute Nörlund summability almost everywhere of orthogonal series (introduction, absolute Nörlund summability of orthogonal series, approximation and absolute summability, structure theorem for trigonometric system, Rademacher trigonometric series, impossibility of improvement of conditions); Chapter 4, Absolute Riesz summability almost everywhere of orthogonal series (introduction, equivalence theorem, sufficient conditions, Rademacher trigonometric series, impossibility of improvement of conditions); Chapter 5, Absolute Nörlund summability factors of Fourier series (introduction, summability factors, some applications, other conditions, some corollaries); Chapter 6, Absolute Nörlund summability factors of conjugate series of Fourier series (introduction, conjugate theorems, some lemmas, proofs of theorems, equivalent relation, some applications); Chapter 7, Local property of absolute Riesz summability of Fourier series (introduction, local property, some …
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 40G05
Mathematics Subject Classification ID: 42A20
zbMATH DE Number: 3857698
zbMATH Keywords: absolute Nörlund summability
zbMATH Keywords: absolute Riesz summability
reviewed by: Ryszard Gajewski
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3857698