Lebesgue Points and Summability of Higher Dimensional Fourier Series

This monograph presents the summability of higher dimensional Fourier series, and generalizes the concept of Lebesgue points. Focusing on Fejér and Cesàro summability, as well as theta-summation, readers will become more familiar with a wide variety of summability methods. Within the theory of higher dimensional summability of Fourier series, the book also provides a much-needed simple proof of Lebesgue’s theorem, filling a gap in the literature. Recent results and real-world applications are highlighted as well, making this a timely resource.The book is structured into four chapters, prioritizing clarity throughout. Chapter One covers basic results from the one-dimensional Fourier series, and offers a clear proof of the Lebesgue theorem. In Chapter Two, convergence and boundedness results for the lq-summability are presented. The restricted and unrestricted rectangular summability are provided in Chapter Three, as well as the sufficient and necessary condition for the norm convergence of the rectangular theta-means. Chapter Four then introduces six types of Lebesgue points for higher dimensional functions.Lebesgue Points and Summability of Higher Dimensional Fourier Series will appeal to researchers working in mathematical analysis, particularly those interested in Fourier and harmonic analysis. Researchers in applied fields will also find this useful.

“The author's aim to make the book almost self-contained is achieved. … The reviewer was a witness to and sometimes a participant in the early days of introducing multidimensional Lebesgue points in the problems of summability of Fourier series and Fourier transforms. It is a pleasure for him, and hopefully one of the advantages of the book, that more or less all the contributions to the topic, even old and not easily accessible ones, are referred to and taken into account.” (Elijah Liyand, Mathematical Reviews, August, 2022)“The book is mainly based on the very many interesting author’s results published in the last 20–30 years. It will be useful for … graduate, postgraduate and Ph.D. students.” (Sorin Gheorghe Gal, zbMATH 1475.42017, 2022)