Lebesgue Points and Summability of Higher Dimensional Fourier Series

Beschreibung

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 l_q-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.

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 l_q-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.

Autor*in

Ferenc Weisz

Themen in »Lebesgue Points and Summability of Higher Dimensional Fourier Series«

Fourier series Fourier transforms higher dimensions Cesaro summability Lebesgue points Fejer summability Fejer summation Lebesgue point theorem Lebesgue theorem Summability Fourier series Circular summability Triangular summability Cubic summability Rectangular summability Hardy-Littlewood maximal functions Theta-summation

Stimmen zu »Lebesgue Points and Summability of Higher Dimensional Fourier Series«

“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)

Details

Link teilen