Functions Spaces And Expansions

Author : Ole Christensen
Publisher : Springer Science & Business Media
Page : 266 pages
File Size : 49,7 Mb
Release : 2010-05-27
Category : Mathematics
ISBN : 9780817649807

Get Book

Functions, Spaces, and Expansions by Ole Christensen Pdf

This graduate-level textbook is a detailed exposition of key mathematical tools in analysis aimed at students, researchers, and practitioners across science and engineering. Every topic covered has been specifically chosen because it plays a key role outside the field of pure mathematics. Although the treatment of each topic is mathematical in nature, and concrete applications are not delineated, the principles and tools presented are fundamental to exploring the computational aspects of physics and engineering. Readers are expected to have a solid understanding of linear algebra, in Rn and in general vector spaces. Familiarity with the basic concepts of calculus and real analysis, including Riemann integrals and infinite series of real or complex numbers, is also required.

Author : W. Trebels
Publisher : Springer
Page : 110 pages
File Size : 53,6 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540469513

Get Book

Multipliers for (C,alpha)-Bounded Fourier Expansions in Banach Spaces and Approximation Theory by W. Trebels Pdf

Author : Robert M Kauffman
Publisher : CRC Press
Page : 158 pages
File Size : 44,8 Mb
Release : 1996-09-25
Category : Mathematics
ISBN : 0582276349

Get Book

Eigenfunction Expansions, Operator Algebras and Riemannian Symmetric Spaces by Robert M Kauffman Pdf

This Research Note pays particular attention to studying the convergence of the expansion and to the case where D is a family of partial differential operators. All operators in the natural von Neumann algebraassociated with D, and also unbounded operators affiliated with this algebra, are expanded simultaneously in terms of generalized eigenprojections. These are operators which carry a natural space associated with D into its dual. The elements of the range of these eigenprojections are the eigenfunctions, which solve the appropriate eigenvalue equations by duality. The spectral measure is abstractly defined, but its absolute continuity with respect to Hausdorf measure on the joint spectrum is shown to occur when the eigenfunctions are very well-behaved. Uniqueness results are given showing that any two expansions arise from each other by a simple change of variable. A considerable effort has been made to keep the book self-contained for readers with a background in functional analysis including a basic understanding of the theory of von Neumann algebras. More advanced topics in functional analysis, andan introduction to differential geometry and differential operator theory, mostly without proofs, are given in an extensive section on background material.

Author : Hans Triebel
Publisher : European Mathematical Society
Page : 276 pages
File Size : 47,9 Mb
Release : 2008
Category : Mathematics
ISBN : 3037190191

Get Book

Function Spaces and Wavelets on Domains by Hans Triebel Pdf

Wavelets have emerged as an important tool in analyzing functions containing discontinuities and sharp spikes. They were developed independently in the fields of mathematics, quantum physics, electrical engineering, and seismic geology. Interchanges between these fields during the last ten years have led to many new wavelet applications such as image compression, turbulence, human vision, radar, earthquake prediction, and pure mathematics applications such as solving partial differential equations. This book develops a theory of wavelet bases and wavelet frames for function spaces on various types of domains. Starting with the usual spaces on Euclidean spaces and their periodic counterparts, the exposition moves on to so-called thick domains (including Lipschitz domains and snowflake domains). Specifically, wavelet expansions and extensions to corresponding spaces on Euclidean $n$-spaces are developed. Finally, spaces on smooth and cellular domains and related manifolds are treated. Although the presentation relies on the recent theory of function spaces, basic notation and classical results are repeated in order to make the text self-contained. This book is addressed to two types of readers: researchers in the theory of function spaces who are interested in wavelets as new effective building blocks for functions and scientists who wish to use wavelet bases in classical function spaces for various applications. Adapted to the second type of reader, the preface contains a guide on where to find basic definitions and key assertions.

Author : Hans Triebel
Publisher : Springer Science & Business Media
Page : 437 pages
File Size : 43,5 Mb
Release : 2012-12-13
Category : Mathematics
ISBN : 9783034805698

Get Book

The Structure of Functions by Hans Triebel Pdf

This book deals with the constructive Weierstrassian approach to the theory of function spaces and various applications. The first chapter is devoted to a detailed study of quarkonial (subatomic) decompositions of functions and distributions on euclidean spaces, domains, manifolds and fractals. This approach combines the advantages of atomic and wavelet representations. It paves the way to sharp inequalities and embeddings in function spaces, spectral theory of fractal elliptic operators, and a regularity theory of some semi-linear equations. The book is self-contained, although some parts may be considered as a continuation of the author's book Fractals and Spectra. It is directed to mathematicians and (theoretical) physicists interested in the topics indicated and, in particular, how they are interrelated. - - - The book under review can be regarded as a continuation of [his book on "Fractals and spectra", 1997] (...) There are many sections named: comments, preparations, motivations, discussions and so on. These parts of the book seem to be very interesting and valuable. They help the reader to deal with the main course. (Mathematical Reviews)

Author : Yutaka Yamamoto
Publisher : SIAM
Page : 282 pages
File Size : 51,7 Mb
Release : 2012-01-01
Category : Mathematics
ISBN : 1611972310

Get Book

From Vector Spaces to Function Spaces by Yutaka Yamamoto Pdf

This book provides a treatment of analytical methods of applied mathematics. It starts with a review of the basics of vector spaces and brings the reader to an advanced discussion of applied mathematics, including the latest applications to systems and control theory. The text is designed to be accessible to those not familiar with the material and useful to working scientists, engineers, and mathematics students. The author provides the motivations of definitions and the ideas underlying proofs but does not sacrifice mathematical rigor. From Vector Spaces to Function Spaces presents: an easily accessible discussion of analytical methods of applied mathematics from vector spaces to distributions, Fourier analysis, and Hardy spaces with applications to system theory; an introduction to modern functional analytic methods to better familiarize readers with basic methods and mathematical thinking; and an understandable yet penetrating treatment of such modern methods and topics as function spaces and distributions, Fourier and Laplace analyses, and Hardy spaces.

Author : Yoshihiro Sawano
Publisher : Springer
Page : 945 pages
File Size : 49,7 Mb
Release : 2018-11-04
Category : Mathematics
ISBN : 9789811308369

Get Book

Theory of Besov Spaces by Yoshihiro Sawano Pdf

This is a self-contained textbook of the theory of Besov spaces and Triebel–Lizorkin spaces oriented toward applications to partial differential equations and problems of harmonic analysis. These include a priori estimates of elliptic differential equations, the T1 theorem, pseudo-differential operators, the generator of semi-group and spaces on domains, and the Kato problem. Various function spaces are introduced to overcome the shortcomings of Besov spaces and Triebel–Lizorkin spaces as well. The only prior knowledge required of readers is familiarity with integration theory and some elementary functional analysis.Illustrations are included to show the complicated way in which spaces are defined. Owing to that complexity, many definitions are required. The necessary terminology is provided at the outset, and the theory of distributions, L^p spaces, the Hardy–Littlewood maximal operator, and the singular integral operators are called upon. One of the highlights is that the proof of the Sobolev embedding theorem is extremely simple. There are two types for each function space: a homogeneous one and an inhomogeneous one. The theory of function spaces, which readers usually learn in a standard course, can be readily applied to the inhomogeneous one. However, that theory is not sufficient for a homogeneous space; it needs to be reinforced with some knowledge of the theory of distributions. This topic, however subtle, is also covered within this volume. Additionally, related function spaces—Hardy spaces, bounded mean oscillation spaces, and Hölder continuous spaces—are defined and discussed, and it is shown that they are special cases of Besov spaces and Triebel–Lizorkin spaces.

Author : Xiaoping Shen,Ahmed I. Zayed
Publisher : Springer Science & Business Media
Page : 378 pages
File Size : 54,7 Mb
Release : 2012-09-18
Category : Technology & Engineering
ISBN : 9781461441458

Get Book

Multiscale Signal Analysis and Modeling by Xiaoping Shen,Ahmed I. Zayed Pdf

Multiscale Signal Analysis and Modeling presents recent advances in multiscale analysis and modeling using wavelets and other systems. This book also presents applications in digital signal processing using sampling theory and techniques from various function spaces, filter design, feature extraction and classification, signal and image representation/transmission, coding, nonparametric statistical signal processing, and statistical learning theory.

Author : M.M. Djrbashian
Publisher : Birkhäuser
Page : 258 pages
File Size : 55,7 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783034885492

Get Book

Harmonic Analysis and Boundary Value Problems in the Complex Domain by M.M. Djrbashian Pdf

As is well known, the first decades of this century were a period of elaboration of new methods in complex analysis. This elaboration had, in particular, one char acteristic feature, consisting in the interfusion of some concepts and methods of harmonic and complex analyses. That interfusion turned out to have great advan tages and gave rise to a vast number of significant results, of which we want to mention especially the classical results on the theory of Fourier series in L2 ( -7r, 7r) and their continual analog - Plancherel's theorem on the Fourier transform in L2 ( -00, +00). We want to note also two important Wiener and Paley theorems on parametric integral representations of a subclass of entire functions of expo nential type in the Hardy space H2 over a half-plane. Being under the strong influence of these results, the author began in the fifties a series of investigations in the theory of integral representations of analytic and entire functions as well as in the theory of harmonic analysis in the com plex domain. These investigations were based on the remarkable properties of the asymptotics of the entire function (p, J1 > 0), which was introduced into mathematical analysis by Mittag-Leffler for the case J1 = 1. In the process of investigation, the scope of some classical results was essentially enlarged, and the results themselves were evaluated.

Author : Karl-Heinz Förster,Peter Jonas,Heinz Langer,Carsten Trunk
Publisher : Springer Science & Business Media
Page : 240 pages
File Size : 44,7 Mb
Release : 2007-12-03
Category : Mathematics
ISBN : 9783764382704

Get Book

Operator Theory in Inner Product Spaces by Karl-Heinz Förster,Peter Jonas,Heinz Langer,Carsten Trunk Pdf

This volume contains contributions written by participants of the 4th Workshop on Operator Theory in Krein Spaces and Applications, held at the TU Berlin, Germany, December 17 to 19, 2004. The workshop covered topics from spectral, perturbation, and extension theory of linear operators and relations in inner product spaces.

Author : Sean Dineen
Publisher : Springer Science & Business Media
Page : 553 pages
File Size : 55,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781447108696

Get Book

Complex Analysis on Infinite Dimensional Spaces by Sean Dineen Pdf

Infinite dimensional holomorphy is the study of holomorphic or analytic func tions over complex topological vector spaces. The terms in this description are easily stated and explained and allow the subject to project itself ini tially, and innocently, as a compact theory with well defined boundaries. However, a comprehensive study would include delving into, and interacting with, not only the obvious topics of topology, several complex variables theory and functional analysis but also, differential geometry, Jordan algebras, Lie groups, operator theory, logic, differential equations and fixed point theory. This diversity leads to a dynamic synthesis of ideas and to an appreciation of a remarkable feature of mathematics - its unity. Unity requires synthesis while synthesis leads to unity. It is necessary to stand back every so often, to take an overall look at one's subject and ask "How has it developed over the last ten, twenty, fifty years? Where is it going? What am I doing?" I was asking these questions during the spring of 1993 as I prepared a short course to be given at Universidade Federal do Rio de Janeiro during the following July. The abundance of suit able material made the selection of topics difficult. For some time I hesitated between two very different aspects of infinite dimensional holomorphy, the geometric-algebraic theory associated with bounded symmetric domains and Jordan triple systems and the topological theory which forms the subject of the present book.

Author : Michael Frazier,Björn Jawerth,Guido L. Weiss
Publisher : American Mathematical Soc.
Page : 132 pages
File Size : 49,5 Mb
Release : 1991
Category : Mathematics
ISBN : 9780821807316

Get Book

Littlewood-Paley Theory and the Study of Function Spaces by Michael Frazier,Björn Jawerth,Guido L. Weiss Pdf

Littlewood-Paley theory was developed to study function spaces in harmonic analysis and partial differential equations. Recently, it has contributed to the development of the $\varphi$-transform and wavelet decompositions. Based on lectures presented at the NSF-CBMS Regional Research Conference on Harmonic Analysis and Function Spaces, held at Auburn University in July 1989, this book is aimed at mathematicians, as well as mathematically literate scientists and engineers interested in harmonic analysis or wavelets. The authors provide not only a general understanding of the area of harmonic analysis relating to Littlewood-Paley theory and atomic and wavelet decompositions, but also some motivation and background helpful in understanding the recent theory of wavelets.The book begins with some simple examples which provide an overview of the classical Littlewood-Paley theory. The $\varphi$-transform, wavelet, and smooth atomic expansions are presented as natural extensions of the classical theory. Finally, applications to harmonic analysis (Calderon-Zygmund operators), signal processing (compression), and mathematical physics (potential theory) are discussed.

Author : J. Nowinski
Publisher : Springer Science & Business Media
Page : 309 pages
File Size : 47,9 Mb
Release : 2013-03-09
Category : Science
ISBN : 9781468439267

Get Book

Applications of Functional Analysis in Engineering by J. Nowinski Pdf

Functional analysis owes its OrIgms to the discovery of certain striking analogies between apparently distinct disciplines of mathematics such as analysis, algebra, and geometry. At the turn of the nineteenth century, a number of observations, made sporadically over the preceding years, began to inspire systematic investigations into the common features of these three disciplines, which have developed rather independently of each other for so long. It was found that many concepts of this triad-analysis, algebra, geometry-could be incorporated into a single, but considerably more abstract, new discipline which came to be called functional analysis. In this way, many aspects of analysis and algebra acquired unexpected and pro found geometric meaning, while geometric methods inspired new lines of approach in analysis and algebra. A first significant step toward the unification and generalization of algebra, analysis, and geometry was taken by Hilbert in 1906, who studied the collection, later called 1 , composed of infinite sequences x = Xb X 2, ... , 2 X , ... , of numbers satisfying the condition that the sum Ik"= 1 X 2 converges. k k The collection 12 became a prototype of the class of collections known today as Hilbert spaces.

Author : A. H. Lightstone,Abraham Robinson
Publisher : Elsevier
Page : 214 pages
File Size : 49,5 Mb
Release : 2016-06-03
Category : Mathematics
ISBN : 9781483257440

Get Book

Nonarchimedean Fields and Asymptotic Expansions by A. H. Lightstone,Abraham Robinson Pdf

North-Holland Mathematical Library, Volume 13: Nonarchimedean Fields and Asymptotic Expansions focuses on the connection between nonarchimedean systems and the orders of infinity and smallness that are related with the asymptotic behavior of a function. The publication first explains nonarchimedean fields and nonstandard analysis. Discussions focus on the method of mathematical logic, ultrapower construction, principles of permanence, internal functions, many-sorted structures, nonarchimedean fields and groups, and fields with evaluation. The text then discusses the Euler-Maclaurin expansions and the formal concept of asymptotic expansions. Topics include a generalized criterion for asymptotic expansions, asymptotic power series, Watson's Lemma, asymptotic sequences, and the Euler-Maclaurin formula. The manuscript examines Popken space, including asymptotically finite functions, convergence, norm, algebraic properties of the norm, and Popken's description of the norm. The text is a dependable reference for mathematicians and researchers interested in nonarchimedean fields and asymptotic expansions.

Author : J.S. Byrnes
Publisher : Springer Science & Business Media
Page : 428 pages
File Size : 46,5 Mb
Release : 2001-09-30
Category : Mathematics
ISBN : 0792371682

Get Book

Twentieth Century Harmonic Analysis by J.S. Byrnes Pdf

Almost a century ago, harmonic analysis entered a (still continuing) Golden Age, with the emergence of many great masters throughout Europe. They created a wealth of profound analytic methods, to be successfully exploited and further developed by succeeding generations. This flourishing of harmonic analysis is today as lively as ever, as the papers presented here demonstrate. In addition to its own ongoing internal development and its basic role in other areas of mathematics, physics and chemistry, financial analysis, medicine, and biological signal processing, harmonic analysis has made fundamental contributions to essentially all twentieth century technology-based human endeavours, including telephone, radio, television, radar, sonar, satellite communications, medical imaging, the Internet, and multimedia. This ubiquitous nature of the subject is amply illustrated. The book not only promotes the infusion of new mathematical tools into applied harmonic analysis, but also to fuel the development of applied mathematics by providing opportunities for young engineers, mathematicians and other scientists to learn more about problem areas in today's technology that might benefit from new mathematical insights.