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Trigonometric Fourier Series and Their Conjugates

  • Book
  • © 1996

Overview

Authors:
  1. Levan Zhizhiashvili

    1. Department of Mechanics and Mathematics, Tbilisi State University, Tbilisi, USA

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Part of the book series: Mathematics and Its Applications (MAIA, volume 372)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xii
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  3. Multiple Trigonometric Series

    1. Front Matter

      Pages 117-117
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    2. Conjugate Functions and Hilbert Transforms of Functions of Several Variables

      • Levan Zhizhiashvili
      Pages 119-166

    3. Convergence and Summability at a Point or Almost Everywhere of Multiple Trigonometric Fourier Series and Their Conjugates

      • Levan Zhizhiashvili
      Pages 167-203

    4. Some Approximating Properties of n-Fold Cesàro Means of the Series \(\sigma _n \left[ {\text{f}} \right]\) and \(\bar \sigma _n \left[ {{\text{f,B}}} \right]\)

      • Levan Zhizhiashvili
      Pages 205-224

    5. Convergence and Summability of Multiple Trigonometric Fourier Series and Their Conjugates in the Spaces LP (Tn), p∞]o,+∞]

      • Levan Zhizhiashvili
      Pages 225-242

    6. Summability of Series \(\sigma _2 \left[ {\text{f}} \right]\) and \(\bar \sigma _2 \left[ {{\text{f,B}}} \right]\) by a Method of the Marcinkiewicz Type

      • Levan Zhizhiashvili
      Pages 243-266

  4. Back Matter

    Pages 267-308
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Keywords

About this book

Research in the theory of trigonometric series has been carried out for over two centuries. The results obtained have greatly influenced various fields of mathematics, mechanics, and physics. Nowadays, the theory of simple trigonometric series has been developed fully enough (we will only mention the monographs by Zygmund [15, 16] and Bari [2]). The achievements in the theory of multiple trigonometric series look rather modest as compared to those in the one-dimensional case though multiple trigonometric series seem to be a natural, interesting and promising object of investigation. We should say, however, that the past few decades have seen a more intensive development of the theory in this field. To form an idea about the theory of multiple trigonometric series, the reader can refer to the surveys by Shapiro [1], Zhizhiashvili [16], [46], Golubov [1], D'yachenko [3]. As to monographs on this topic, only that ofYanushauskas [1] is known to me. This book covers several aspects of the theory of multiple trigonometric Fourier series: the existence and properties of the conjugates and Hilbert transforms of integrable functions; convergence (pointwise and in the LP-norm, p > 0) of Fourier series and their conjugates, as well as their summability by the Cesaro (C,a), a> -1, and Abel-Poisson methods; approximating properties of Cesaro means of Fourier series and their conjugates.

Reviews

` ... wealth of material collected ... We recommend it to everyone who wants to get fresh information on recent results in a very traditional part of Fourier analysis.'
Mathematical Reviews Clippings 97k

Authors and Affiliations

  • Department of Mechanics and Mathematics, Tbilisi State University, Tbilisi, USA

    Levan Zhizhiashvili

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