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. Preface Contents List of Figures 1 One-Dimensional Fourier Series 1.1 The Lp Spaces 1.2 Convergence of Fourier Series 1.3 Hardy-Littlewood Maximal Function and Lebesgue Points 1.4 Summability of One-Dimensional Fourier Series 1.5 Convergence at Lebesgue Points of the Cesàro Means 2 ellq-Summability of Higher Dimensional Fourier Series 2.1 Higher Dimensional Partial Sums 2.2 The ellq-Summability Kernels 2.2.1 Kernel Functions for q=1 2.2.2 Kernel Functions for q=infty 2.2.3 Kernel Functions for q=2 2.3 Norm Convergence of the ellq-Summability Means 2.4 Hp(mathbbTd) Hardy Spaces 2.5 Almost Everywhere Convergence of the ellq-Summability Means 2.5.1 Almost Everywhere Convergence for q=1 and q=infty 2.5.2 Almost Everywhere Convergence for q=2 2.6 ellq-Summability Defined by a Function θ 2.6.1 Triangular and Cubic Summability 2.6.2 Circular Summability 2.6.3 Some Summability Methods 3 Rectangular Summability of Higher Dimensional Fourier Series 3.1 Summability Kernels 3.2 Norm Convergence of Rectangular Summability Means 3.3 Almost Everywhere Restricted Summability over a Cone 3.4 Almost Everywhere Restricted Summability over a Cone-Like Set 3.5 Hp(mathbbTd) Hardy spaces 3.6 Almost Everywhere Unrestricted Summability 3.7 Rectangular θ-Summability 3.7.1 Feichtinger's Algebra S0(mathbbRd) 3.7.2 Norm Convergence of the Rectangular θ-Means 3.7.3 Almost Everywhere Convergence of the Rectangular θ-Means 3.7.4 Some Summability Methods 4 Lebesgue Points of Higher Dimensional Functions 4.1 ell2-Summability 4.1.1 Hardy-Littlewood Maximal Functions 4.1.2 Lebesgue Points for the ell2-Summability 4.2 Unrestricted Rectangular Summability 4.2.1 Strong Hardy-Littlewood Maximal Functions 4.2.2 Lebesgue Points for the Unrestricted Rectangular Summability 4.2.3 Some Applications 4.3 Restricted Rectangular Summability over a Cone 4.3.1 Hardy-Littlewood Maximal Functions 4.3.2 Lebesgue Points for the Summability over a Cone 4.4 Restricted Rectangular Summability over a Cone-Like Set 4.4.1 Hardy-Littlewood Maximal Functions 4.4.2 Lebesgue Points for the Summability over a Cone-Like Set 4.5 ellinfty-Summability 4.5.1 Hardy-Littlewood Maximal Functions 4.5.2 Lebesgue Points for the ellinfty-Summability 4.6 ell1-Summability Appendix Bibliography Index Notations Lake Index