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1996 | Buch

The Conjugation Operator and the Hilbert Transform Kapitel lesen Erstes Kapitel lesen

Trigonometric Fourier Series and Their Conjugates

verfasst von: Levan Zhizhiashvili

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Simple Trigonometric Series

Frontmatter
Chapter I. The Conjugation Operator and the Hilbert Transform
Abstract
Throughout the book we use the notation \( T = \left[ { - \pi ,\pi } \right],\mathbb{R} = \left] { - \infty , + \infty } \right[, \) and \( \mathbb{R}_ + = \left[ {0, + \infty } \right[. \) Given \( p \in \left] {0, + \infty } \right[,L^p \left( T \right) \) will stand for the set of all 2π-periodic measurable functions f:ℝ→ℝ for which the expression
$$ \left\| f \right\|_p = \left\{ {\frac{1} {{2\pi }}\int\limits_T {\left| {f\left( x \right)} \right|^p dx} } \right\}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}} $$
is finite, while for p = ∞, we will assume
$$ L^\infty \left( T \right) = C\left( T \right),\left\| f \right\|_\infty = \mathop {\sup }\limits_{x \in T} \left| {f\left( x \right)} \right|. $$
Levan Zhizhiashvili
Chapter II. Pointwise Convergence and Summability of Trigonometric Series
Abstract
Let
$$ \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {a_k \cos kx + b_k \sin kx} $$
(2.1.1)
be a trigonometric series and
$$ \sum\limits_{k = 1}^\infty {b_k \cos kx - a_k \sin kx} $$
(2.1.2)
be its conjugate. We will denote by \( S_n \left( x \right) \) and \( \bar S_n \left( x \right) \) the partial sums of (2.1.1) and (2.1.2), respectively.
Levan Zhizhiashvili
Chapter III. Convergence and Summability of Trigonometric Fourier Series and Their Conjugates in the Spaces
Abstract
In 1925, Kolmogorov [3] showed that if \( f \in L\left( T \right), \) then for any \( p \in \left] {0,1} \right[, \) it holds
$$ \left\| {S_n \left( f \right) - f} \right\|_p \to 0,\left\| {\bar S_n \left( f \right) - \bar f} \right\|_p \to 0asn \to \infty . $$
(3.1.1)
Levan Zhizhiashvili
Chapter IV. Some Approximating Properties of Cesàro Means of the Series and
Abstract
By now, a great deal of fundamental and profound results has been established concerning approximating properties of means of Cesaro, Poisson-Abel, Hardy, and of general linear means of the series \( \sigma \left[ f \right] \) and \( \bar \sigma \left[ f \right] \). Most of them are set forth in monographs and proceedings of international conferences, schools and symposia. Even several monographs would not suffice for a complete account of the results obtained in this direction. Therefore below we will survey only those of them which are necessary in the study of the questions treated in this chapter. We note, however, that the monographs by Alexits [2], Akhiezer [1], Bari [2], Vallée-Poussin [1], Jackson [3], Dzyadyk [1], Zygmund [15], Korneïchuk [1], Leindler [1], Lorentz [2], Natanson [1], Nikol’skiï [5], Stepanets [1], Timan [1], as well as the papers by Korneïchuk [2] and Tikhomirov [1], give the reader an idea of the most important investigations inthis field.
Levan Zhizhiashvili

Multiple Trigonometric Series

Frontmatter
Chapter I. Conjugate Functions and Hilbert Transforms of Functions of Several Variables
Abstract
The theory of multiple trigonometric and orthogonal series is rapidly developing today. The results are set down in a number of monographs and surveys. We mention only a part of them: Alimov, Il’in, Nikishin [1], [2], Alimov, Ashurov and Pulatov [1], Golubov [1], D’yachenko [3], Dyn’kin [1], [2], Dyn’kin and Osilenker [1], Zhizhiashvili [16], [35], [46], Zygmund [16], Kislyakov [2], Stein [3], Stein and Weiss [1], Suetin [1], Topuria [1], Torchinsky [1], Khavin [2], Shapiro [1], Yanushauskas [1].
Levan Zhizhiashvili
Chapter II. Convergence and Summability at a Point or Almost Everywhere of Multiple Trigonometric Fourier Series and Their Conjugates
Abstract
In the works [13, Ch. IV], [16], [35, Ch. II], we have set down various questions connected with the convergence and summability at a point or almost everywhere of multiple trigomometric series and their conjugates. Some other results in this direction are presented in our works [15], [24], [37], [39], [41], [42], [45].
Levan Zhizhiashvili
Chapter III. Some Approximating Properties of n-Fold Cesàro Means of the Series and
Abstract
In Chapter 4 of the first part, we have presented some results connected with approximating properties of Cesàro means of the series \(\sigma \left[ f \right]\) and \(\bar \sigma \left[ f \right]\). In this chapter, their multidimensional analogues are given. We have treated these items in [4], [5], [13, Ch. IV], [35], [46].
Levan Zhizhiashvili
Chapter IV. Convergence and Summability of Multiple Trigonometric Fourier Series and Their Conjugates in the Spaces LP (Tn), p∞]o,+∞]
Abstract
In 1925, Kolmogorov [3] for the first time used the method of representation of partial sums of the series σ[f] and σ[f] in terms of conjugate functions, and established the convergence of the series σ[/] in L P (T) for any p ∞ [0,1[. Later on, this method was widely used in the works of various mathemeticians (see, e.g., Zygmund [15, Ch. VII], Lozinskiï [1], etc.). In particular, for partial sums of σ[f] and σ[f], there hold (in appropriate Lebesgue classes) the inequalities of types of Kolmogorov [3] (see (1; 1.1.21)) and Zygmund [4] (see (1;1.1.5)), [15, Ch. VII] (see (1.1.47)).
Levan Zhizhiashvili
Chapter V. Summability of Series and by a Method of the Marcinkiewicz Type
Abstract
This chapter basically deals with the case n = 2.
Levan Zhizhiashvili
Backmatter
Metadaten
Titel
Trigonometric Fourier Series and Their Conjugates
verfasst von
Levan Zhizhiashvili
Copyright-Jahr
1996
Verlag
Springer Netherlands
Electronic ISBN
978-94-009-0283-1
Print ISBN
978-94-010-6612-9
DOI
https://doi.org/10.1007/978-94-009-0283-1