Bochner-Riesz Means on Euclidean Spaces PDF Download

Author: Shanzhen Lu
Publisher: World Scientific
ISBN: 9814458775
Category : Mathematics
Languages : en
Pages : 385

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Book Description
This book mainly deals with the BochnerOCoRiesz means of multiple Fourier integral and series on Euclidean spaces. It aims to give a systematical introduction to the fundamental theories of the BochnerOCoRiesz means and important achievements attained in the last 50 years. For the BochnerOCoRiesz means of multiple Fourier integral, it includes the Fefferman theorem which negates the Disc multiplier conjecture, the famous Carleson-SjAlin theorem, and Carbery-Rubio de Francia-Vega's work on almost everywhere convergence of the BochnerOCoRiesz means below the critical index. For the BochnerOCoRiesz means of multiple Fourier series, it includes the theory and application of a class of function space generated by blocks, which is closely related to almost everywhere convergence of the BochnerOCoRiesz means. In addition, the book also introduce some research results on approximation of functions by the BochnerOCoRiesz means.

Author: Ferenc Weisz
Publisher: Springer Nature
ISBN: 3030746364
Category : Mathematics
Languages : en
Pages : 299

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Book Description
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.

Author: Marco Annoni
Publisher:
ISBN:
Category : Bochner technique
Languages : en
Pages : 171

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Book Description
The Fourier transform is a mathematical operation that can be used with its inverse to rewrite a function as a sum of waves. It has been a useful mathematical tool for many applied sciences. Sometimes Fourier inversion is not possible in the classic sense and needs to be generalized. This is often done in a standard way, after choosing a summability method. A famous and much studied one is the method of the Bochner-Riesz means. We use techniques and results of harmonic analysis (Plancharel-type inequalities, partitions of the euclidean space, an analytic continuation argument, maximal operators, duality, potentials etc.) to investigate the method of the Bochner-Riesz means modified by A. Seeger. We prove that the Fourier inversion with respect to the modified Bochner-Riesz means holds pointwise almost everywhere for a certain class of functions. First of all, this result refines a Theorem of A. Carbery, J. Rubio de Francia and L. Vega. Secondly, it highlights the connection between the choice of the method one can use to invert the Fourier transform and the class of functions on which the method will work. Finally, it also shows how to generalize certain techniques to a scenario where we lack certain algebraic properties.

Author: Ferenc Weisz
Publisher: Birkhäuser
ISBN: 3319568140
Category : Mathematics
Languages : en
Pages : 446

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Book Description
This book investigates the convergence and summability of both one-dimensional and multi-dimensional Fourier transforms, as well as the theory of Hardy spaces. To do so, it studies a general summability method known as theta-summation, which encompasses all the well-known summability methods, such as the Fejér, Riesz, Weierstrass, Abel, Picard, Bessel and Rogosinski summations. Following on the classic books by Bary (1964) and Zygmund (1968), this is the first book that considers strong summability introduced by current methodology. A further unique aspect is that the Lebesgue points are also studied in the theory of multi-dimensional summability. In addition to classical results, results from the past 20-30 years – normally only found in scattered research papers – are also gathered and discussed, offering readers a convenient “one-stop” source to support their work. As such, the book will be useful for researchers, graduate and postgraduate students alike.

Author: Katherine Michelle Davis
Publisher: Cambridge University Press
ISBN: 9780521312776
Category : Mathematics
Languages : en
Pages : 168

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Book Description
Beginning with a thorough discussion of the classical one-dimensional theory, this text considers the modern theory of Fourier series since Zygmund's classic study. It covers developments of the 1970s from Fefferman's famous disc counterexample to Cordoba's geometric theory.

Author: Shanzhen Lu
Publisher: World Scientific
ISBN: 9811253692
Category : Mathematics
Languages : en
Pages : 215

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Book Description
In many branches of mathematical analysis and mathematical physics, the Hardy operator and Hardy inequality are fundamentally important and have been intensively studied ever since the pioneer researches. This volume presents new properties of higher-dimensional Hardy operators obtained by the authors and their collaborators over the last decade. Its prime focus is on higher-dimensional Hardy operators that are based on the spherical average form.The key motivation for this monograph is based on the fact that the Hardy operator is generally smaller than the Hardy-Littlewood maximal operator, which leads to, on the one hand, the operator norm of the Hardy operator itself being smaller than the latter. On the other hand, the former characterizing the weight function class or function spaces is greater than the latter.

Author: Pankaj Jain
Publisher: Springer
ISBN: 981106119X
Category : Mathematics
Languages : en
Pages : 334

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Book Description
This book features original research and survey articles on the topics of function spaces and inequalities. It focuses on (variable/grand/small) Lebesgue spaces, Orlicz spaces, Lorentz spaces, and Morrey spaces and deals with mapping properties of operators, (weighted) inequalities, pointwise multipliers and interpolation. Moreover, it considers Sobolev–Besov and Triebel–Lizorkin type smoothness spaces. The book includes papers by leading international researchers, presented at the International Conference on Function Spaces and Inequalities, held at the South Asian University, New Delhi, India, on 11–15 December 2015, which focused on recent developments in the theory of spaces with variable exponents. It also offers further investigations concerning Sobolev-type embeddings, discrete inequalities and harmonic analysis. Each chapter is dedicated to a specific topic and written by leading experts, providing an overview of the subject and stimulating future research.

Author: Massimo A. Picardello
Publisher: Springer Science & Business Media
ISBN: 8847028531
Category : Mathematics
Languages : en
Pages : 450

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Book Description
This book illustrates the wide range of research subjects developed by the Italian research group in harmonic analysis, originally started by Alessandro Figà-Talamanca, to whom it is dedicated in the occasion of his retirement. In particular, it outlines some of the impressive ramifications of the mathematical developments that began when Figà-Talamanca brought the study of harmonic analysis to Italy; the research group that he nurtured has now expanded to cover many areas. Therefore the book is addressed not only to experts in harmonic analysis, summability of Fourier series and singular integrals, but also in potential theory, symmetric spaces, analysis and partial differential equations on Riemannian manifolds, analysis on graphs, trees, buildings and discrete groups, Lie groups and Lie algebras, and even in far-reaching applications as for instance cellular automata and signal processing (low-discrepancy sampling, Gaussian noise).

Author: Loukas Grafakos
Publisher: Springer Science & Business Media
ISBN: 0387094342
Category : Mathematics
Languages : en
Pages : 517

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Book Description
The great response to the publication of the book Classical and Modern Fourier Analysishasbeenverygratifying.IamdelightedthatSpringerhasofferedtopublish the second edition of this book in two volumes: Classical Fourier Analysis, 2nd Edition, and Modern Fourier Analysis, 2nd Edition. These volumes are mainly addressed to graduate students who wish to study Fourier analysis. This second volume is intended to serve as a text for a seco- semester course in the subject. It is designed to be a continuation of the rst v- ume. Chapters 1–5 in the rst volume contain Lebesgue spaces, Lorentz spaces and interpolation, maximal functions, Fourier transforms and distributions, an introd- tion to Fourier analysis on the n-torus, singular integrals of convolution type, and Littlewood–Paley theory. Armed with the knowledgeof this material, in this volume,the reader encounters more advanced topics in Fourier analysis whose development has led to important theorems. These theorems are proved in great detail and their proofs are organized to present the ow of ideas. The exercises at the end of each section enrich the material of the corresponding section and provide an opportunity to develop ad- tional intuition and deeper comprehension. The historical notes in each chapter are intended to provide an account of past research but also to suggest directions for further investigation. The auxiliary results referred to the appendix can be located in the rst volume.

Author: Óscar Domínguez
Publisher: American Mathematical Society
ISBN: 1470455382
Category : Mathematics
Languages : en
Pages : 180

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Book Description
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