Harmonic Function Theory
Harmonic Function Theory

Author: Sheldon Axler

Publisher: Springer Science & Business Media

Published: 2013-11-11

Total Pages: 266

ISBN-13: 1475781377

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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.

Positive Harmonic Functions and Diffusion
Positive Harmonic Functions and Diffusion

Author: Ross G. Pinsky

Publisher: Cambridge University Press

Published: 1995-01-12

Total Pages: 492

ISBN-13: 0521470145

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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.

Harmonic Functions and Potentials on Finite or Infinite Networks
Harmonic Functions and Potentials on Finite or Infinite Networks

Author: Victor Anandam

Publisher: Springer Science & Business Media

Published: 2011-06-27

Total Pages: 152

ISBN-13: 3642213995

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Random walks, Markov chains and electrical networks serve as an introduction to the study of real-valued functions on finite or infinite graphs, with appropriate interpretations using probability theory and current-voltage laws. The relation between this type of function theory and the (Newton) potential theory on the Euclidean spaces is well-established. The latter theory has been variously generalized, one example being the axiomatic potential theory on locally compact spaces developed by Brelot, with later ramifications from Bauer, Constantinescu and Cornea. A network is a graph with edge-weights that need not be symmetric. This book presents an autonomous theory of harmonic functions and potentials defined on a finite or infinite network, on the lines of axiomatic potential theory. Random walks and electrical networks are important sources for the advancement of the theory.

Harmonic Analysis, Differential Equations, Calculus of Variations with Application to Stability of Complex Geometric Structures
Harmonic Analysis, Differential Equations, Calculus of Variations with Application to Stability of Complex Geometric Structures

Author: E. M. Stein

Publisher:

Published: 1971

Total Pages: 5

ISBN-13:

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The report summarizes the research results obtained under the referenced Grant during the period 1 July 68 through 30 June 71, lists the manuscripts and reprints which report these results, and lists the individuals (partially) supported under the Grant. Briefly stated, these results concern the convergence of Poisson integrals, the analogues of singular integral operators and certain pseudo-differential operators, the Littlewood-Paley theory, Markov chains, the Neumann problem, the boundary behavior of positive harmonic functions, integral transforms and singular integrals. (Author).

Probabilistic Behavior of Harmonic Functions
Probabilistic Behavior of Harmonic Functions

Author: Rodrigo Banuelos

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 220

ISBN-13: 3034887280

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Harmonic analysis and probability have long enjoyed a mutually beneficial relationship that has been rich and fruitful. This monograph, aimed at researchers and students in these fields, explores several aspects of this relationship. The primary focus of the text is the nontangential maximal function and the area function of a harmonic function and their probabilistic analogues in martingale theory. The text first gives the requisite background material from harmonic analysis and discusses known results concerning the nontangential maximal function and area function, as well as the central and essential role these have played in the development of the field.The book next discusses further refinements of traditional results: among these are sharp good-lambda inequalities and laws of the iterated logarithm involving nontangential maximal functions and area functions. Many applications of these results are given. Throughout, the constant interplay between probability and harmonic analysis is emphasized and explained. The text contains some new and many recent results combined in a coherent presentation.

Subharmonic Functions
Subharmonic Functions

Author: W. K. Hayman

Publisher: Elsevier

Published: 2014-06-28

Total Pages: 618

ISBN-13: 1483296180

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Building on the foundation laid in the first volume of Subharmonic Functions, which has become a classic, this second volume deals extensively with applications to functions of a complex variable. The material also has applications in differential equations and differential equations and differential geometry. It reflects the increasingly important role that subharmonic functions play in these areas of mathematics. The presentation goes back to the pioneering work of Ahlfors, Heins, and Kjellberg, leading to and including the more recent results of Baernstein, Weitsman, and many others. The volume also includes some previously unpublished material. It addresses mathematicians from graduate students to researchers in the field and will also appeal to physicists and electrical engineers who use these tools in their research work. The extensive preface and introductions to each chapter give readers an overview. A series of examples helps readers test their understatnding of the theory and the master the applications.