Lebesgue Points And Summability Of Higher Dimensional Fourier Series

Author : Ferenc Weisz
Publisher : Springer Nature
Page : 299 pages
File Size : 42,7 Mb
Release : 2021-06-12
Category : Mathematics
ISBN : 9783030746360

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Lebesgue Points and Summability of Higher Dimensional Fourier Series by Ferenc Weisz Pdf

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 : Ferenc Weisz
Publisher : Birkhäuser
Page : 435 pages
File Size : 50,7 Mb
Release : 2017-12-27
Category : Mathematics
ISBN : 9783319568140

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Convergence and Summability of Fourier Transforms and Hardy Spaces by Ferenc Weisz Pdf

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 : Lars-Erik Persson,George Tephnadze,Ferenc Weisz
Publisher : Springer Nature
Page : 633 pages
File Size : 49,7 Mb
Release : 2022-11-22
Category : Mathematics
ISBN : 9783031144592

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Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series by Lars-Erik Persson,George Tephnadze,Ferenc Weisz Pdf

This book discusses, develops and applies the theory of Vilenkin-Fourier series connected to modern harmonic analysis. The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. Such waves have already been used frequently in the theory of signal transmission, multiplexing, filtering, image enhancement, code theory, digital signal processing and pattern recognition. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series. Because of this it is inevitable to compare results of Vilenkin-Fourier series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group. The first part of the book can be used as an introduction to the subject, and the following chapters summarize the most recent research in this fascinating area and can be read independently. Each chapter concludes with historical remarks and open questions. The book will appeal to researchers working in Fourier and more broad harmonic analysis and will inspire them for their own and their students' research. Moreover, researchers in applied fields will appreciate it as a sourcebook far beyond the traditional mathematical domains.

Author : Yinqin Li,Dachun Yang,Long Huang
Publisher : Springer Nature
Page : 663 pages
File Size : 41,6 Mb
Release : 2023-02-14
Category : Mathematics
ISBN : 9789811967887

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Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko by Yinqin Li,Dachun Yang,Long Huang Pdf

The real-variable theory of function spaces has always been at the core of harmonic analysis. In particular, the real-variable theory of the Hardy space is a fundamental tool of harmonic analysis, with applications and connections to complex analysis, partial differential equations, and functional analysis. This book is devoted to exploring properties of generalized Herz spaces and establishing a complete real-variable theory of Hardy spaces associated with local and global generalized Herz spaces via a totally fresh perspective. This means that the authors view these generalized Herz spaces as special cases of ball quasi-Banach function spaces. In this book, the authors first give some basic properties of generalized Herz spaces and obtain the boundedness and the compactness characterizations of commutators on them. Then the authors introduce the associated Herz–Hardy spaces, localized Herz–Hardy spaces, and weak Herz–Hardy spaces, and develop a complete real-variable theory of these Herz–Hardy spaces, including their various maximal function, atomic, molecular as well as various Littlewood–Paley function characterizations. As applications, the authors establish the boundedness of some important operators arising from harmonic analysis on these Herz–Hardy spaces. Finally, the inhomogeneous Herz–Hardy spaces and their complete real-variable theory are also investigated. With the fresh perspective and the improved conclusions on the real-variable theory of Hardy spaces associated with ball quasi-Banach function spaces, all the obtained results of this book are new and their related exponents are sharp. This book will be appealing to researchers and graduate students who are interested in function spaces and their applications.

Author : Ferenc Weisz
Publisher : Springer Science & Business Media
Page : 340 pages
File Size : 48,5 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9789401731836

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Summability of Multi-Dimensional Fourier Series and Hardy Spaces by Ferenc Weisz Pdf

The history of martingale theory goes back to the early fifties when Doob [57] pointed out the connection between martingales and analytic functions. On the basis of Burkholder's scientific achievements the mar tingale theory can perfectly well be applied in complex analysis and in the theory of classical Hardy spaces. This connection is the main point of Durrett's book [60]. The martingale theory can also be well applied in stochastics and mathematical finance. The theories of the one-parameter martingale and the classical Hardy spaces are discussed exhaustively in the literature (see Garsia [83], Neveu [138], Dellacherie and Meyer [54, 55], Long [124], Weisz [216] and Duren [59], Stein [193, 194], Stein and Weiss [192], Lu [125], Uchiyama [205]). The theory of more-parameter martingales and martingale Hardy spaces is investigated in Imkeller [107] and Weisz [216]. This is the first mono graph which considers the theory of more-parameter classical Hardy spaces. The methods of proofs for one and several parameters are en tirely different; in most cases the theorems stated for several parameters are much more difficult to verify. The so-called atomic decomposition method that can be applied both in the one-and more-parameter cases, was considered for martingales by the author in [216].

Author : Juan Arias de Reyna
Publisher : Springer Science & Business Media
Page : 210 pages
File Size : 41,9 Mb
Release : 2002-04
Category : Mathematics
ISBN : 3540432701

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Pointwise Convergence of Fourier Series by Juan Arias de Reyna Pdf

This book contains a detailed exposition of Carleson-Hunt theorem following the proof of Carleson: to this day this is the only one giving better bounds. It points out the motivation of every step in the proof. Thus the Carleson-Hunt theorem becomes accessible to any analyst.The book also contains the first detailed exposition of the fine results of Hunt, Sjölin, Soria, etc on the convergence of Fourier Series. Its final chapters present original material. With both Fefferman's proof and the recent one of Lacey and Thiele in print, it becomes more important than ever to understand and compare these two related proofs with that of Carleson and Hunt. These alternative proofs do not yield all the results of the Carleson-Hunt proof. The intention of this monograph is to make Carleson's proof accessible to a wider audience, and to explain its consequences for the pointwise convergence of Fourier series for functions in spaces near $äcal Lü^1$, filling a well-known gap in the literature.

Author : Aubrey Henderson Smith
Publisher : Unknown
Page : 26 pages
File Size : 46,7 Mb
Release : 1933
Category : Fourier series
ISBN : PSU:000003571125

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On the Summability of Derived Series of the Fourier-Lebesgue Type by Aubrey Henderson Smith Pdf

Author : Thomas Walsh
Publisher : American Mathematical Soc.
Page : 103 pages
File Size : 55,8 Mb
Release : 1973
Category : Fonctions de plusieurs variables réelles
ISBN : 9780821818312

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On Summability Methods for Conjugate Fourier-Stieltjes Integrals in Several Variables and Generalizations by Thomas Walsh Pdf

Author : Michael Thoreau Lacey,Xiaochun Li
Publisher : American Mathematical Soc.
Page : 87 pages
File Size : 54,5 Mb
Release : 2010
Category : Harmonic analysis
ISBN : 9780821845400

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On a Conjecture of E. M. Stein on the Hilbert Transform on Vector Fields by Michael Thoreau Lacey,Xiaochun Li Pdf

"Volume 205, number 965 (fourth of 5 numbers)."

Author : Igor Iakovlevič Novikov (mathématicien).),Vladimir I︠U︡rʹevich Protasov,Marii︠a︡ Aleksandrovna Skopina
Publisher : American Mathematical Soc.
Page : 522 pages
File Size : 49,9 Mb
Release : 2011
Category : Mathematics
ISBN : 9780821849842

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Wavelet Theory by Igor Iakovlevič Novikov (mathématicien).),Vladimir I︠U︡rʹevich Protasov,Marii︠a︡ Aleksandrovna Skopina Pdf

Wavelet theory lies on the crossroad of pure and computational mathematics, with connections to audio and video signal processing, data compression, and information transmission. The present book is devoted to a systematic exposition of modern wavelet theory. It details the construction of orthogonal and biorthogonal systems of wavelets and studies their structural and approximation properties, starting with basic theory and ending with special topics and problems. The book also presents some applications of wavelets. Historical commentary is supplied for each chapter in the book, and most chapters contain exercises. The book is intended for professional mathematicians and graduate students working in functional analysis and approximation theory. It is also useful for engineers applying wavelet theory in their work. Prerequisites for reading the book consist of graduate courses in real and functional analysis.

Author : Marianna Bolla,Tamás Szabados
Publisher : CRC Press
Page : 318 pages
File Size : 45,9 Mb
Release : 2021-04-29
Category : Mathematics
ISBN : 9781000392395

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Multidimensional Stationary Time Series by Marianna Bolla,Tamás Szabados Pdf

This book gives a brief survey of the theory of multidimensional (multivariate), weakly stationary time series, with emphasis on dimension reduction and prediction. Understanding the covered material requires a certain mathematical maturity, a degree of knowledge in probability theory, linear algebra, and also in real, complex and functional analysis. For this, the cited literature and the Appendix contain all necessary material. The main tools of the book include harmonic analysis, some abstract algebra, and state space methods: linear time-invariant filters, factorization of rational spectral densities, and methods that reduce the rank of the spectral density matrix. Serves to find analogies between classical results (Cramer, Wold, Kolmogorov, Wiener, Kálmán, Rozanov) and up-to-date methods for dimension reduction in multidimensional time series Provides a unified treatment for time and frequency domain inferences by using machinery of complex and harmonic analysis, spectral and Smith--McMillan decompositions. Establishes analogies between the time and frequency domain notions and calculations Discusses the Wold's decomposition and the Kolmogorov's classification together, by distinguishing between different types of singularities. Understanding the remote past helps us to characterize the ideal situation where there is a regular part at present. Examples and constructions are also given Establishes a common outline structure for the state space models, prediction, and innovation algorithms with unified notions and principles, which is applicable to real-life high frequency time series It is an ideal companion for graduate students studying the theory of multivariate time series and researchers working in this field.

Author : Loukas Grafakos
Publisher : Springer
Page : 647 pages
File Size : 51,6 Mb
Release : 2014-11-17
Category : Mathematics
ISBN : 9781493911943

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Classical Fourier Analysis by Loukas Grafakos Pdf

The main goal of this text is to present the theoretical foundation of the field of Fourier analysis on Euclidean spaces. It covers classical topics such as interpolation, Fourier series, the Fourier transform, maximal functions, singular integrals, and Littlewood–Paley theory. The primary readership is intended to be graduate students in mathematics with the prerequisite including satisfactory completion of courses in real and complex variables. The coverage of topics and exposition style are designed to leave no gaps in understanding and stimulate further study. This third edition includes new Sections 3.5, 4.4, 4.5 as well as a new chapter on “Weighted Inequalities,” which has been moved from GTM 250, 2nd Edition. Appendices I and B.9 are also new to this edition. Countless corrections and improvements have been made to the material from the second edition. Additions and improvements include: more examples and applications, new and more relevant hints for the existing exercises, new exercises, and improved references.

Author : Terence Tao
Publisher : American Mathematical Soc.
Page : 206 pages
File Size : 41,7 Mb
Release : 2021-09-03
Category : Education
ISBN : 9781470466404

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An Introduction to Measure Theory by Terence Tao Pdf

This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Carathéodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.

Author : Richard Tolimieri,Myoung An,Chao Lu
Publisher : Unknown
Page : 256 pages
File Size : 51,6 Mb
Release : 1993
Category : Mathematics
ISBN : UOM:39015029094839

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Mathematics of Multidimensional Fourier Transform Alogrithms by Richard Tolimieri,Myoung An,Chao Lu Pdf

Author : Kenneth L. Cooke
Publisher : Unknown
Page : 158 pages
File Size : 48,5 Mb
Release : 1948
Category : Fourier series
ISBN : STANFORD:36105011917114

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The Convergence and Summability of Fourier Serico by Kenneth L. Cooke Pdf