Harmonic Spaces

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Potential Theory on Harmonic Spaces

Author : Corneliu Constantinescu
Publisher : Springer
ISBN 13 : 9783642654343
Total Pages : 0 pages
Book Rating : 4.6/5 (543 download)

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Book Synopsis Potential Theory on Harmonic Spaces by : Corneliu Constantinescu

Download or read book Potential Theory on Harmonic Spaces written by Corneliu Constantinescu and published by Springer. This book was released on 2012-01-16 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: There has been a considerable revival of interest in potential theory during the last 20 years. This is made evident by the appearance of new mathematical disciplines in that period which now-a-days are considered as parts of potential theory. Examples of such disciplines are: the theory of Choquet capacities, of Dirichlet spaces, of martingales and Markov processes, of integral representation in convex compact sets as well as the theory of harmonic spaces. All these theories have roots in classical potential theory. The theory of harmonic spaces, sometimes also called axiomatic theory of harmonic functions, plays a particular role among the above mentioned theories. On the one hand, this theory has particularly close connections with classical potential theory. Its main notion is that of a harmonic function and its main aim is the generalization and unification of classical results and methods for application to an extended class of elliptic and parabolic second order partial differential equations. On the other hand, the theory of harmonic spaces is closely related to the theory of Markov processes. In fact, all important notions and results of the theory have a probabilistic interpretation.

Harmonic Analysis on Spaces of Homogeneous Type

Author : Donggao Deng
Publisher : Springer Science & Business Media
ISBN 13 : 354088744X
Total Pages : 167 pages
Book Rating : 4.5/5 (48 download)

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Book Synopsis Harmonic Analysis on Spaces of Homogeneous Type by : Donggao Deng

Download or read book Harmonic Analysis on Spaces of Homogeneous Type written by Donggao Deng and published by Springer Science & Business Media. This book was released on 2008-11-19 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book could have been entitled “Analysis and Geometry.” The authors are addressing the following issue: Is it possible to perform some harmonic analysis on a set? Harmonic analysis on groups has a long tradition. Here we are given a metric set X with a (positive) Borel measure ? and we would like to construct some algorithms which in the classical setting rely on the Fourier transformation. Needless to say, the Fourier transformation does not exist on an arbitrary metric set. This endeavor is not a revolution. It is a continuation of a line of research whichwasinitiated,acenturyago,withtwofundamentalpapersthatIwould like to discuss brie?y. The ?rst paper is the doctoral dissertation of Alfred Haar, which was submitted at to University of Gottingen ̈ in July 1907. At that time it was known that the Fourier series expansion of a continuous function may diverge at a given point. Haar wanted to know if this phenomenon happens for every 2 orthonormal basis of L [0,1]. He answered this question by constructing an orthonormal basis (today known as the Haar basis) with the property that the expansion (in this basis) of any continuous function uniformly converges to that function.

Harmonic Spaces

Author : Harold Stanley Ruse
Publisher :
ISBN 13 :
Total Pages : 264 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Harmonic Spaces by : Harold Stanley Ruse

Download or read book Harmonic Spaces written by Harold Stanley Ruse and published by . This book was released on 1961 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Variable Lebesgue Spaces

Author : David V. Cruz-Uribe
Publisher : Springer Science & Business Media
ISBN 13 : 3034805489
Total Pages : 316 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Variable Lebesgue Spaces by : David V. Cruz-Uribe

Download or read book Variable Lebesgue Spaces written by David V. Cruz-Uribe and published by Springer Science & Business Media. This book was released on 2013-02-12 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an accessible introduction to the theory of variable Lebesgue spaces. These spaces generalize the classical Lebesgue spaces by replacing the constant exponent p with a variable exponent p(x). They were introduced in the early 1930s but have become the focus of renewed interest since the early 1990s because of their connection with the calculus of variations and partial differential equations with nonstandard growth conditions, and for their applications to problems in physics and image processing. The book begins with the development of the basic function space properties. It avoids a more abstract, functional analysis approach, instead emphasizing an hands-on approach that makes clear the similarities and differences between the variable and classical Lebesgue spaces. The subsequent chapters are devoted to harmonic analysis on variable Lebesgue spaces. The theory of the Hardy-Littlewood maximal operator is completely developed, and the connections between variable Lebesgue spaces and the weighted norm inequalities are introduced. The other important operators in harmonic analysis - singular integrals, Riesz potentials, and approximate identities - are treated using a powerful generalization of the Rubio de Francia theory of extrapolation from the theory of weighted norm inequalities. The final chapter applies the results from previous chapters to prove basic results about variable Sobolev spaces.

Harmonic Analysis on Homogeneous Spaces

Author : Nolan R. Wallach
Publisher : Courier Dover Publications
ISBN 13 : 0486816923
Total Pages : 386 pages
Book Rating : 4.4/5 (868 download)

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Book Synopsis Harmonic Analysis on Homogeneous Spaces by : Nolan R. Wallach

Download or read book Harmonic Analysis on Homogeneous Spaces written by Nolan R. Wallach and published by Courier Dover Publications. This book was released on 2018-12-18 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is suitable for advanced undergraduate and graduate students in mathematics with a strong background in linear algebra and advanced calculus. Early chapters develop representation theory of compact Lie groups with applications to topology, geometry, and analysis, including the Peter-Weyl theorem, the theorem of the highest weight, the character theory, invariant differential operators on homogeneous vector bundles, and Bott's index theorem for such operators. Later chapters study the structure of representation theory and analysis of non-compact semi-simple Lie groups, including the principal series, intertwining operators, asymptotics of matrix coefficients, and an important special case of the Plancherel theorem. Teachers will find this volume useful as either a main text or a supplement to standard one-year courses in Lie groups and Lie algebras. The treatment advances from fairly simple topics to more complex subjects, and exercises appear at the end of each chapter. Eight helpful Appendixes develop aspects of differential geometry, Lie theory, and functional analysis employed in the main text.

Harmonic Analysis on Commutative Spaces

Author : Joseph Albert Wolf
Publisher : American Mathematical Soc.
ISBN 13 : 0821842897
Total Pages : 408 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Harmonic Analysis on Commutative Spaces by : Joseph Albert Wolf

Download or read book Harmonic Analysis on Commutative Spaces written by Joseph Albert Wolf and published by American Mathematical Soc.. This book was released on 2007 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: This study starts with the basic theory of topological groups, harmonic analysis, and unitary representations. It then concentrates on geometric structure, harmonic analysis, and unitary representation theory in commutative spaces.

Dirichlet Integrals on Harmonic Spaces

Author : F.-Y. Maeda
Publisher : Springer
ISBN 13 : 3540393013
Total Pages : 190 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Dirichlet Integrals on Harmonic Spaces by : F.-Y. Maeda

Download or read book Dirichlet Integrals on Harmonic Spaces written by F.-Y. Maeda and published by Springer. This book was released on 2006-11-14 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces

Author : Jürgen Berndt
Publisher : Springer
ISBN 13 : 3540491716
Total Pages : 135 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces by : Jürgen Berndt

Download or read book Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces written by Jürgen Berndt and published by Springer. This book was released on 2006-11-14 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: Generalized Heisenberg groups, or H-type groups, introduced by A. Kaplan, and Damek-Ricci harmonic spaces are particularly nice Lie groups with a vast spectrum of properties and applications. These harmonic spaces are homogeneous Hadamard manifolds containing the H-type groups as horospheres. These notes contain a thorough study of their Riemannian geometry by means of a detailed treatment of their Jacobi vector fields and Jacobi operators. Some problems are included and will hopefully stimulate further research on these spaces. The book is written for students and researchers, assuming only basic knowledge of Riemannian geometry, and it contains a brief survey of the background material needed to follow the entire treatment.

Groups Acting on Hyperbolic Space

Author : Juergen Elstrodt
Publisher : Springer Science & Business Media
ISBN 13 : 3662036266
Total Pages : 530 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Groups Acting on Hyperbolic Space by : Juergen Elstrodt

Download or read book Groups Acting on Hyperbolic Space written by Juergen Elstrodt and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 530 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is concerned with discontinuous groups of motions of the unique connected and simply connected Riemannian 3-manifold of constant curva ture -1, which is traditionally called hyperbolic 3-space. This space is the 3-dimensional instance of an analogous Riemannian manifold which exists uniquely in every dimension n :::: 2. The hyperbolic spaces appeared first in the work of Lobachevski in the first half of the 19th century. Very early in the last century the group of isometries of these spaces was studied by Steiner, when he looked at the group generated by the inversions in spheres. The ge ometries underlying the hyperbolic spaces were of fundamental importance since Lobachevski, Bolyai and Gauß had observed that they do not satisfy the axiom of parallels. Already in the classical works several concrete coordinate models of hy perbolic 3-space have appeared. They make explicit computations possible and also give identifications of the full group of motions or isometries with well-known matrix groups. One such model, due to H. Poincare, is the upper 3 half-space IH in JR . The group of isometries is then identified with an exten sion of index 2 of the group PSL(2,

Harmonic Analysis on Symmetric Spaces and Applications I

Author : Audrey Terras
Publisher : Springer
ISBN 13 :
Total Pages : 368 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Harmonic Analysis on Symmetric Spaces and Applications I by : Audrey Terras

Download or read book Harmonic Analysis on Symmetric Spaces and Applications I written by Audrey Terras and published by Springer. This book was released on 1985-07 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since its beginnings with Fourier (and as far back as the Babylonian astron omers), harmonic analysis has been developed with the goal of unraveling the mysteries of the physical world of quasars, brain tumors, and so forth, as well as the mysteries of the nonphysical, but no less concrete, world of prime numbers, diophantine equations, and zeta functions. Quoting Courant and Hilbert, in the preface to the first German edition of Methods of Mathematical Physics: "Recent trends and fashions have, however, weakened the connection between mathematics and physics." Such trends are still in evidence, harmful though they may be. My main motivation in writing these notes has been a desire to counteract this tendency towards specialization and describe appli cations of harmonic analysis in such diverse areas as number theory (which happens to be my specialty), statistics, medicine, geophysics, and quantum physics. I remember being quite surprised to learn that the subject is useful. My graduate education was that of the 1960s. The standard mathematics graduate course proceeded from Definition 1. 1. 1 to Corollary 14. 5. 59, with no room in between for applications, motivation, history, or references to related work. My aim has been to write a set of notes for a very different sort of course.

Harmonic Analysis and Special Functions on Symmetric Spaces

Author : Gerrit Heckman
Publisher : Academic Press
ISBN 13 : 0080533299
Total Pages : 239 pages
Book Rating : 4.0/5 (85 download)

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Book Synopsis Harmonic Analysis and Special Functions on Symmetric Spaces by : Gerrit Heckman

Download or read book Harmonic Analysis and Special Functions on Symmetric Spaces written by Gerrit Heckman and published by Academic Press. This book was released on 1995-02-08 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: The two parts of this sharply focused book, Hypergeometric and Special Functions and Harmonic Analysis on Semisimple Symmetric Spaces, are derived from lecture notes for the European School of Group Theory, a forum providing high-level courses on recent developments in group theory. The authors provide students and researchers with a thorough and thoughtful overview, elaborating on the topic with clear statements of definitions and theorems and augmenting these withtime-saving examples. An extensive set of notes supplements the text. Heckman and Schlichtkrull extend the ideas of harmonic analysis on semisimple symmetric spaces to embrace the theory of hypergeometric and spherical functions and show that the K-variant Eisenstein integrals for G/H are hypergeometric functions under this theory. They lead readers from the fundamentals of semisimple symmetric spaces of G/H to the frontier, including generalization, to the Riemannian case. This volume will interest harmonic analysts, those working on or applying the theory of symmetric spaces; it will also appeal to those with an interest in special functions. Extends ideas of harmonic analysis on symmetric spaces First treatment of the theory to include hypergeometric and spherical functions Links algebraic, analytic, and geometric methods

Harmonic Analysis of Operators on Hilbert Space

Author : Béla Sz Nagy
Publisher : Springer Science & Business Media
ISBN 13 : 1441960937
Total Pages : 481 pages
Book Rating : 4.4/5 (419 download)

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Book Synopsis Harmonic Analysis of Operators on Hilbert Space by : Béla Sz Nagy

Download or read book Harmonic Analysis of Operators on Hilbert Space written by Béla Sz Nagy and published by Springer Science & Business Media. This book was released on 2010-09-01 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: The existence of unitary dilations makes it possible to study arbitrary contractions on a Hilbert space using the tools of harmonic analysis. The first edition of this book was an account of the progress done in this direction in 1950-70. Since then, this work has influenced many other areas of mathematics, most notably interpolation theory and control theory. This second edition, in addition to revising and amending the original text, focuses on further developments of the theory, including the study of two operator classes: operators whose powers do not converge strongly to zero, and operators whose functional calculus (as introduced in Chapter III) is not injective. For both of these classes, a wealth of material on structure, classification and invariant subspaces is included in Chapters IX and X. Several chapters conclude with a sketch of other developments related with (and developing) the material of the first edition.

Harmonic Analysis in Phase Space. (AM-122), Volume 122

Author : Gerald B. Folland
Publisher : Princeton University Press
ISBN 13 : 1400882427
Total Pages : 288 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Harmonic Analysis in Phase Space. (AM-122), Volume 122 by : Gerald B. Folland

Download or read book Harmonic Analysis in Phase Space. (AM-122), Volume 122 written by Gerald B. Folland and published by Princeton University Press. This book was released on 2016-03-02 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides the first coherent account of the area of analysis that involves the Heisenberg group, quantization, the Weyl calculus, the metaplectic representation, wave packets, and related concepts. This circle of ideas comes principally from mathematical physics, partial differential equations, and Fourier analysis, and it illuminates all these subjects. The principal features of the book are as follows: a thorough treatment of the representations of the Heisenberg group, their associated integral transforms, and the metaplectic representation; an exposition of the Weyl calculus of pseudodifferential operators, with emphasis on ideas coming from harmonic analysis and physics; a discussion of wave packet transforms and their applications; and a new development of Howe's theory of the oscillator semigroup.

Banach Spaces, Harmonic Analysis, and Probability Theory

Author : R. C. Blei
Publisher : Springer
ISBN 13 : 3540400362
Total Pages : 183 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Banach Spaces, Harmonic Analysis, and Probability Theory by : R. C. Blei

Download or read book Banach Spaces, Harmonic Analysis, and Probability Theory written by R. C. Blei and published by Springer. This book was released on 2006-11-15 with total page 183 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Martingale Theory in Harmonic Analysis and Banach Spaces

Author : J.-A. Chao
Publisher : Springer
ISBN 13 : 354039284X
Total Pages : 238 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Martingale Theory in Harmonic Analysis and Banach Spaces by : J.-A. Chao

Download or read book Martingale Theory in Harmonic Analysis and Banach Spaces written by J.-A. Chao and published by Springer. This book was released on 2006-11-17 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Harmonic Function Theory

Author : Sheldon Axler
Publisher : Springer Science & Business Media
ISBN 13 : 1475781377
Total Pages : 266 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Harmonic Function Theory by : Sheldon Axler

Download or read book Harmonic Function Theory written by Sheldon Axler and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is about harmonic functions in Euclidean space. This new edition contains a completely rewritten chapter on spherical harmonics, a new section on extensions of Bochers Theorem, new exercises and proofs, as well as revisions throughout to improve the text. A unique software package supplements the text for readers who wish to explore harmonic function theory on a computer.

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Author : Audrey Terras
Publisher : Springer Science & Business Media
ISBN 13 : 146147972X
Total Pages : 430 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane by : Audrey Terras

Download or read book Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane written by Audrey Terras and published by Springer Science & Business Media. This book was released on 2013-09-12 with total page 430 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré upper half plane. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Written with an informal style, the book places an emphasis on motivation, concrete examples, history, and, above all, applications in mathematics, statistics, physics, and engineering. Many corrections and updates have been incorporated in this new edition. Updates include discussions of P. Sarnak and others' work on quantum chaos, the work of T. Sunada, Marie-France Vignéras, Carolyn Gordon, and others on Mark Kac's question "Can you hear the shape of a drum?", A. Lubotzky, R. Phillips and P. Sarnak's examples of Ramanujan graphs, and, finally, the author's comparisons of continuous theory with the finite analogues. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, Poisson's summation formula and applications in crystallography and number theory, applications of spherical harmonic analysis to the hydrogen atom, the Radon transform, non-Euclidean geometry on the Poincaré upper half plane H or unit disc and applications to microwave engineering, fundamental domains in H for discrete groups Γ, tessellations of H from such discrete group actions, automorphic forms, and the Selberg trace formula and its applications in spectral theory as well as number theory.