# Trigonometric Fourier Series and Their Conjugates

• Levan Zhizhiashvili
Book

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

1. Front Matter
Pages i-xii
2. ### Simple Trigonometric Series

1. Front Matter
Pages 1-1
2. Levan Zhizhiashvili
Pages 3-33
3. Levan Zhizhiashvili
Pages 35-70
4. Levan Zhizhiashvili
Pages 93-116
3. ### Multiple Trigonometric Series

1. Front Matter
Pages 117-117
2. Levan Zhizhiashvili
Pages 119-166
3. Levan Zhizhiashvili
Pages 167-203
4. Levan Zhizhiashvili
Pages 225-242
4. Back Matter
Pages 267-308

### Introduction

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.

### Keywords

calculus fourier analysis integral transform operational calculus

#### Authors and affiliations

• Levan Zhizhiashvili
• 1
1. 1.Department of Mechanics and MathematicsTbilisi State UniversityTbilisiUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-94-009-0283-1