Convergence And Summability Of Fourier Transforms And Hardy Spaces
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Convergence And Summability Of Fourier Transforms And Hardy Spaces
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Author : Ferenc Weisz
language : en
Publisher: Birkhäuser
Release Date : 2017-12-27
Convergence And Summability Of Fourier Transforms And Hardy Spaces written by Ferenc Weisz and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-12-27 with Mathematics categories.
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.
Martingale Hardy Spaces And Summability Of One Dimensional Vilenkin Fourier Series
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Author : Lars-Erik Persson
language : en
Publisher: Springer Nature
Release Date : 2022-11-22
Martingale Hardy Spaces And Summability Of One Dimensional Vilenkin Fourier Series written by Lars-Erik Persson and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-11-22 with Mathematics categories.
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.
Lebesgue Points And Summability Of Higher Dimensional Fourier Series
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Author : Ferenc Weisz
language : en
Publisher: Springer Nature
Release Date : 2021-06-12
Lebesgue Points And Summability Of Higher Dimensional Fourier Series written by Ferenc Weisz and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-12 with Mathematics categories.
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.
Functions Of Bounded Variation And Their Fourier Transforms
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Author : Elijah Liflyand
language : en
Publisher: Springer
Release Date : 2019-03-06
Functions Of Bounded Variation And Their Fourier Transforms written by Elijah Liflyand and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-03-06 with Mathematics categories.
Functions of bounded variation represent an important class of functions. Studying their Fourier transforms is a valuable means of revealing their analytic properties. Moreover, it brings to light new interrelations between these functions and the real Hardy space and, correspondingly, between the Fourier transform and the Hilbert transform. This book is divided into two major parts, the first of which addresses several aspects of the behavior of the Fourier transform of a function of bounded variation in dimension one. In turn, the second part examines the Fourier transforms of multivariate functions with bounded Hardy variation. The results obtained are subsequently applicable to problems in approximation theory, summability of the Fourier series and integrability of trigonometric series.
Summability Of Multi Dimensional Fourier Series And Hardy Spaces
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Author : Ferenc Weisz
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
Summability Of Multi Dimensional Fourier Series And Hardy Spaces written by Ferenc Weisz and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-29 with Mathematics categories.
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].
Real Variable Theory Of Hardy Spaces Associated With Generalized Herz Spaces Of Rafeiro And Samko
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Author : Yinqin Li
language : en
Publisher: Springer Nature
Release Date : 2023-02-14
Real Variable Theory Of Hardy Spaces Associated With Generalized Herz Spaces Of Rafeiro And Samko written by Yinqin Li and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-02-14 with Mathematics categories.
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.
Analysis Of Divergence
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Author : William Bray
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Analysis Of Divergence written by William Bray and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
The 7th International Workshop in Analysis and its Applications (IWAA) was held at the University of Maine, June 1-6, 1997 and featured approxi mately 60 mathematicians. The principal theme of the workshop shares the title of this volume and the latter is a direct outgrowth of the workshop. IWAA was founded in 1984 by Professor Caslav V. Stanojevic. The first meeting was held in the resort complex Kupuri, Yugoslavia, June 1-10, 1986, with two pilot meetings preceding. The Organization Committee to gether with the Advisory Committee (R. P. Boas, R. R. Goldberg, J. P. Kahne) set forward the format and content of future meetings. A certain number of papers were presented that later appeared individually in such journals as the Proceedings of the AMS, Bulletin of the AMS, Mathematis chen Annalen, and the Journal of Mathematical Analysis and its Applica tions. The second meeting took place June 1-10, 1987, at the same location. At the plenary session of this meeting it was decided that future meetings should have a principal theme. The theme for the third meeting (June 1- 10, 1989, Kupuri) was Karamata's Regular Variation. The principal theme for the fourth meeting (June 1-10, 1990, Kupuri) was Inner Product and Convexity Structures in Analysis, Mathematical Physics, and Economics. The fifth meeting was to have had the theme, Analysis and Foundations, organized in cooperation with Professor A. Blass (June 1-10, 1991, Kupuri).
Course In Analysis A Vol Iv Fourier Analysis Ordinary Differential Equations Calculus Of Variations
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Author : Niels Jacob
language : en
Publisher: World Scientific
Release Date : 2018-07-19
Course In Analysis A Vol Iv Fourier Analysis Ordinary Differential Equations Calculus Of Variations written by Niels Jacob and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-07-19 with Mathematics categories.
In the part on Fourier analysis, we discuss pointwise convergence results, summability methods and, of course, convergence in the quadratic mean of Fourier series. More advanced topics include a first discussion of Hardy spaces. We also spend some time handling general orthogonal series expansions, in particular, related to orthogonal polynomials. Then we switch to the Fourier integral, i.e. the Fourier transform in Schwartz space, as well as in some Lebesgue spaces or of measures.Our treatment of ordinary differential equations starts with a discussion of some classical methods to obtain explicit integrals, followed by the existence theorems of Picard-Lindelöf and Peano which are proved by fixed point arguments. Linear systems are treated in great detail and we start a first discussion on boundary value problems. In particular, we look at Sturm-Liouville problems and orthogonal expansions. We also handle the hypergeometric differential equations (using complex methods) and their relations to special functions in mathematical physics. Some qualitative aspects are treated too, e.g. stability results (Ljapunov functions), phase diagrams, or flows.Our introduction to the calculus of variations includes a discussion of the Euler-Lagrange equations, the Legendre theory of necessary and sufficient conditions, and aspects of the Hamilton-Jacobi theory. Related first order partial differential equations are treated in more detail.The text serves as a companion to lecture courses, and it is also suitable for self-study. The text is complemented by ca. 260 problems with detailed solutions.
A Course In Analysis
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Author : Niels Jacob
language : en
Publisher: World Scientific Publishing Company
Release Date : 2016
A Course In Analysis written by Niels Jacob and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016 with Calculus categories.
In the part on Fourier analysis, we discuss pointwise convergence results, summability methods and, of course, convergence in the quadratic mean of Fourier series. More advanced topics include a first discussion of Hardy spaces. We also spend some time handling general orthogonal series expansions, in particular, related to orthogonal polynomials. Then we switch to the Fourier integral, i.e. the Fourier transform in Schwartz space, as well as in some Lebesgue spaces or of measures. Our treatment of ordinary differential equations starts with a discussion of some classical methods to obtain explicit integrals, followed by the existence theorems of Picard-Lindelöf and Peano which are proved by fixed point arguments. Linear systems are treated in great detail and we start a first discussion on boundary value problems. In particular, we look at Sturm-Liouville problems and orthogonal expansions. We also handle the hypergeometric differential equations (using complex methods) and their relations to special functions in mathematical physics. Some qualitative aspects are treated too, e.g. stability results (Ljapunov functions), phase diagrams, or flows. Our introduction to the calculus of variations includes a discussion of the Euler-Lagrange equations, the Legendre theory of necessary and sufficient conditions, and aspects of the Hamilton-Jacobi theory. Related first order partial differential equations are treated in more detail. The text serves as a companion to lecture courses, and it is also suitable for self-study. The text is complemented by ca. 260 problems with detailed solutions.
Contributions To Fourier Analysis
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Author : Antoni Zygmund
language : en
Publisher: Princeton University Press
Release Date : 1950-08-21
Contributions To Fourier Analysis written by Antoni Zygmund and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1950-08-21 with Mathematics categories.
A classic treatment of Fourier analysis from the acclaimed Annals of Mathematics Studies series Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century. To mark the continued success of the series, all books are available in paperback and as ebooks.