Trigonometric Fourier Series and Their Conjugates
(Sprache: Englisch)
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...
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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 ofthe 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.
This book presents in a coherent way the results obtained in the following aspects of the theory of multiple trigonometric Fourier series: the existence and properties of the conjugates and Hilbert transforms of integrable functions of several variables; convergence of Fourier series and their conjugates, as well as their summability by Cesàro and Abel-Poisson methods; and approximating properties of Cesàro means of Fourier series and their conjugates.
Special emphasis is put on new effects which arise from dealing with multiple series and which are not inherent in the one-dimensional case. Unsolved problems are formulated separately.
Audience: This volume will prove useful to both graduate students and research workers in the field of Fourier analysis, approximations and expansions, integral transforms, and operational calculus.
Special emphasis is put on new effects which arise from dealing with multiple series and which are not inherent in the one-dimensional case. Unsolved problems are formulated separately.
Audience: This volume will prove useful to both graduate students and research workers in the field of Fourier analysis, approximations and expansions, integral transforms, and operational calculus.
Inhaltsverzeichnis zu „Trigonometric Fourier Series and Their Conjugates “
Preface. Part 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 Lp(T), p epsilon]0,+INFINITY[. IV. Some Approximating Properties of Cesàro Means of the Series sigma[f] and sigma-bar[f]. Part 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 Cesàro Means of the Series sigman[f] and sigma-barn[f,B]. IV. Convergence and Summability of Multiple Trigonometric Fourier Series and Their Conjugates in the Spaces Lp(Tn), p epsilon]0,+INFINITY]. V. Summability of Series sigma2[f] and sigma-bar2[f,B]. Bibliography. Index.
Bibliographische Angaben
- Autor: L. Zhizhiashvili
- 1996, 308 Seiten, Maße: 23,5 cm, Gebunden, Englisch
- Verlag: Springer Netherlands
- ISBN-10: 0792340884
- ISBN-13: 9780792340881
Sprache:
Englisch
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