Author : Sheldon Axler
Publisher : Springer Science & Business Media
Page : 266 pages
File Size : 36,87 MB
Release : 2013-11-11
Category : Mathematics
ISBN : 1475781377
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.
Author : Sheldon Axler
Publisher : Springer
Page : 238 pages
File Size : 32,96 MB
Release : 2006-05-04
Category : Mathematics
ISBN : 0387215271
Harmonic functions - the solutions of Laplace's equation - play a crucial role in many areas of mathematics, physics, and engineering. Avoiding the disorganization and inconsistent notation of other expositions, the authors approach the field from a more function-theoretic perspective, emphasizing techniques and results that will seem natural to mathematicians comfortable with complex function theory and harmonic analysis; prerequisites for the book are a solid foundation in real and complex analysis together with some basic results from functional analysis. Topics covered include: basic properties of harmonic functions defined on subsets of Rn, including Poisson integrals; properties bounded functions and positive functions, including Liouville's and Cauchy's theorems; the Kelvin transform; Spherical harmonics; hp theory on the unit ball and on half-spaces; harmonic Bergman spaces; the decomposition theorem; Laurent expansions and classification of isolated singularities; and boundary behavior. An appendix describes routines for use with MATHEMATICA to manipulate some of the expressions that arise in the study of harmonic functions.
Author : Sheldon Axler
Publisher : Springer Science & Business Media
Page : 262 pages
File Size : 36,35 MB
Release : 2001-01-25
Category : Mathematics
ISBN : 0387952187
This is a book about harmonic functions in Euclidean space. Readers with a background in real and complex analysis at the beginning graduate level will feel comfortable with the material presented here. The authors have taken unusual care to motivate concepts and simplify proofs. Topics include: basic properties of harmonic functions, Poisson integrals, the Kelvin transform, spherical harmonics, harmonic Hardy spaces, harmonic Bergman spaces, the decomposition theorem, Laurent expansions, isolated singularities, and the Dirichlet problem. The new edition contains a completely rewritten chapter on spherical harmonics, a new section on extensions of Bocher's Theorem, new exercises and proofs, as well as revisions throughout to improve the text. A unique software package-designed by the authors and available by e-mail - supplements the text for readers who wish to explore harmonic function theory on a computer.
Author : Daniel Harrison
Publisher : University of Chicago Press
Page : 364 pages
File Size : 39,9 MB
Release : 1994-05-28
Category : Music
ISBN : 9780226318080
Applicable on a wide scale not only to this repertory, Harrison's lucid explications of abstract theoretical concepts provide new insights into the workings of tonal systems in general.
Author : R.E. Greene
Publisher : Springer
Page : 219 pages
File Size : 18,14 MB
Release : 2006-11-15
Category : Mathematics
ISBN : 3540355367
Author : Steven G. Krantz
Publisher : Springer Science & Business Media
Page : 367 pages
File Size : 25,40 MB
Release : 2009-05-24
Category : Mathematics
ISBN : 0817646698
This self-contained text provides an introduction to modern harmonic analysis in the context in which it is actually applied, in particular, through complex function theory and partial differential equations. It takes the novice mathematical reader from the rudiments of harmonic analysis (Fourier series) to the Fourier transform, pseudodifferential operators, and finally to Heisenberg analysis.
Author : Ross G. Pinsky
Publisher : Cambridge University Press
Page : 492 pages
File Size : 36,33 MB
Release : 1995-01-12
Category : Mathematics
ISBN : 0521470145
In this book, Professor Pinsky gives a self-contained account of the theory of positive harmonic functions for second order elliptic operators, using an integrated probabilistic and analytic approach. The book begins with a treatment of the construction and basic properties of diffusion processes. This theory then serves as a vehicle for studying positive harmonic funtions. Starting with a rigorous treatment of the spectral theory of elliptic operators with nice coefficients on smooth, bounded domains, the author then develops the theory of the generalized principal eigenvalue, and the related criticality theory for elliptic operators on arbitrary domains. Martin boundary theory is considered, and the Martin boundary is explicitly calculated for several classes of operators. The book provides an array of criteria for determining whether a diffusion process is transient or recurrent. Also introduced are the theory of bounded harmonic functions, and Brownian motion on manifolds of negative curvature. Many results that form the folklore of the subject are here given a rigorous exposition, making this book a useful reference for the specialist, and an excellent guide for the graduate student.
Author : Steven George Krantz
Publisher : American Mathematical Soc.
Page : 586 pages
File Size : 26,55 MB
Release : 2001
Category : Mathematics
ISBN : 0821827243
Emphasizing integral formulas, the geometric theory of pseudoconvexity, estimates, partial differential equations, approximation theory, inner functions, invariant metrics, and mapping theory, this title is intended for the student with a background in real and complex variable theory, harmonic analysis, and differential equations.
Author : Cho-Ho Chu
Publisher : Springer
Page : 113 pages
File Size : 36,59 MB
Release : 2004-10-11
Category : Mathematics
ISBN : 3540477934
This research monograph introduces some new aspects to the theory of harmonic functions and related topics. The authors study the analytic algebraic structures of the space of bounded harmonic functions on locally compact groups and its non-commutative analogue, the space of harmonic functionals on Fourier algebras. Both spaces are shown to be the range of a contractive projection on a von Neumann algebra and therefore admit Jordan algebraic structures. This provides a natural setting to apply recent results from non-associative analysis, semigroups and Fourier algebras. Topics discussed include Poisson representations, Poisson spaces, quotients of Fourier algebras and the Murray-von Neumann classification of harmonic functionals.
Author : Corneliu Constantinescu
Publisher : Springer
Page : 376 pages
File Size : 38,30 MB
Release : 1972-12-05
Category : Mathematics
ISBN :
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.
