Trigonometric Fourier Series and Their Conjugates

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.

Zusammenfassung
` ... 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.'
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Inhalt
Preface.- 1 Simple Trigonometric Series.- I. The Conjugation Operator and the Hilbert Transform.- II. Pointwise Convergence and Summability of Trigonometric Series.- III. Convergence and Summability of Trigonometric Fourier Series and Their Conjugates in the Spaces $$L^p \left( T \right),p \in \left] {0, + \infty } \right[$$.- IV. Some Approximating Properties of Cesaro Means of the Series $$ \sigma \left[ f \right] $$ and $$ \bar \sigma \left[ f \right] $$.- 2 Multiple Trigonometric Series.- I. Conjugate Functions and Hilbert Transforms of Functions of Several Variables.- II. Convergence and Summability at a Point or Almost Everywhere of Multiple Trigonometric Fourier Series and Their Conjugates.- III. Some Approximating Properties of n-Fold Cesaro Means of the Series $$ \sigma _n \left[ f \right] $$ and $$ \sigma _n \left[ {f,B} \right] $$.- IV. Convergence and Summability of Multiple Trigonometric Fourier Series and Their Conjugates in the Spaces $$ L^p \left( {T^n } \right),p \in \left] {0, + \infty } \right] $$.- V. Summability of Series $$ \sigma _2 \left[ f \right] $$ and $$ \bar \sigma _2 \left[ {f,B} \right] $$ by a Method of the Marcinkiewicz Type.