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Author: F. Di Biase Publisher: Springer Science & Business Media ISBN: 1461223105 Category : Mathematics Languages : en Pages : 158
Book Description
A basic principle governing the boundary behaviour of holomorphic func tions (and harmonic functions) is this: Under certain growth conditions, for almost every point in the boundary of the domain, these functions ad mit a boundary limit, if we approach the bounda-ry point within certain approach regions. For example, for bounded harmonic functions in the open unit disc, the natural approach regions are nontangential triangles with one vertex in the boundary point, and entirely contained in the disc [Fat06]. In fact, these natural approach regions are optimal, in the sense that convergence will fail if we approach the boundary inside larger regions, having a higher order of contact with the boundary. The first theorem of this sort is due to J. E. Littlewood [Lit27], who proved that if we replace a nontangential region with the rotates of any fixed tangential curve, then convergence fails. In 1984, A. Nagel and E. M. Stein proved that in Euclidean half spaces (and the unit disc) there are in effect regions of convergence that are not nontangential: These larger approach regions contain tangential sequences (as opposed to tangential curves). The phenomenon discovered by Nagel and Stein indicates that the boundary behaviour of ho)omor phic functions (and harmonic functions), in theorems of Fatou type, is regulated by a second principle, which predicts the existence of regions of convergence that are sequentially larger than the natural ones.
Author: F. Di Biase Publisher: Springer Science & Business Media ISBN: 1461223105 Category : Mathematics Languages : en Pages : 158
Book Description
A basic principle governing the boundary behaviour of holomorphic func tions (and harmonic functions) is this: Under certain growth conditions, for almost every point in the boundary of the domain, these functions ad mit a boundary limit, if we approach the bounda-ry point within certain approach regions. For example, for bounded harmonic functions in the open unit disc, the natural approach regions are nontangential triangles with one vertex in the boundary point, and entirely contained in the disc [Fat06]. In fact, these natural approach regions are optimal, in the sense that convergence will fail if we approach the boundary inside larger regions, having a higher order of contact with the boundary. The first theorem of this sort is due to J. E. Littlewood [Lit27], who proved that if we replace a nontangential region with the rotates of any fixed tangential curve, then convergence fails. In 1984, A. Nagel and E. M. Stein proved that in Euclidean half spaces (and the unit disc) there are in effect regions of convergence that are not nontangential: These larger approach regions contain tangential sequences (as opposed to tangential curves). The phenomenon discovered by Nagel and Stein indicates that the boundary behaviour of ho)omor phic functions (and harmonic functions), in theorems of Fatou type, is regulated by a second principle, which predicts the existence of regions of convergence that are sequentially larger than the natural ones.
Author: J. L. Doob Publisher: Springer Science & Business Media ISBN: 1461252083 Category : Mathematics Languages : en Pages : 865
Book Description
Potential theory and certain aspects of probability theory are intimately related, perhaps most obviously in that the transition function determining a Markov process can be used to define the Green function of a potential theory. Thus it is possible to define and develop many potential theoretic concepts probabilistically, a procedure potential theorists observe withjaun diced eyes in view of the fact that now as in the past their subject provides the motivation for much of Markov process theory. However that may be it is clear that certain concepts in potential theory correspond closely to concepts in probability theory, specifically to concepts in martingale theory. For example, superharmonic functions correspond to supermartingales. More specifically: the Fatou type boundary limit theorems in potential theory correspond to supermartingale convergence theorems; the limit properties of monotone sequences of superharmonic functions correspond surprisingly closely to limit properties of monotone sequences of super martingales; certain positive superharmonic functions [supermartingales] are called "potentials," have associated measures in their respective theories and are subject to domination principles (inequalities) involving the supports of those measures; in each theory there is a reduction operation whose properties are the same in the two theories and these reductions induce sweeping (balayage) of the measures associated with potentials, and so on.
Author: Jacob Korevaar Publisher: Springer Science & Business Media ISBN: 3662102250 Category : Mathematics Languages : en Pages : 497
Book Description
Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates. The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early results. He shows the fascination of the difficult Hardy-Littlewood theorems and of an unexpected simple proof, and extolls Wiener's breakthrough based on Fourier theory. There are the spectacular "high-indices" theorems and Karamata's "regular variation", which permeates probability theory. The author presents Gelfand's elegant algebraic treatment of Wiener theory and his own distributional approach. There is also a new unified theory for Borel and "circle" methods. The text describes many Tauberian ways to the prime number theorem. A large bibliography and a substantial index round out the book.
Author: Jordanka Paneva-konovska Publisher: World Scientific ISBN: 1786340909 Category : Mathematics Languages : en Pages : 228
Book Description
Bessel and Mittag-Leffler functions are prominent within mathematical and scientific fields due to increasing interest in non-conventional models within applied mathematics. Since the analytical solutions of many differential and integral equations of arbitrary order can be written as series of special functions of fractional calculus, they are now unavoidable tools for handling various mathematical models of integer or fractional order. From Bessel to Multi-Index Mittag-Leffler Functions analyzes this through the study of enumerable families of different classes of special functions.Enumerable families are considered and the convergence of series is investigated. Providing a unified approach to the classical power series, analogues of the classical results for the power series are obtained, and the conclusion is that each of the considered series has a similar convergence behavior to a power series. Also studied are various properties of the Bessel and Mittag-Leffler functions and their generalizations, including estimations, asymptotic formulae, fractional differentiation and integration operators.
Author: Dorina Mitrea Publisher: Springer Nature ISBN: 3031227352 Category : Mathematics Languages : en Pages : 980
Book Description
This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations. Volume III is concerned with integral representation formulas for nullsolutions of elliptic PDEs, Calderón-Zygmund theory for singular integral operators, Fatou type theorems for systems of elliptic PDEs, and applications to acoustic and electromagnetic scattering. Overall, this amounts to a powerful and nuanced theory developed on uniformly rectifiable sets, which builds on the work of many predecessors.
Author: Adam Korǹyi (et al.) Publisher: American Mathematical Soc. ISBN: 082180605X Category : Mathematics Languages : en Pages : 194
Book Description
This volume presents the proceedings of the workshop "Harmonic Functions on Graphs" held at the Graduate Centre of CUNY in the autumn of 1995. The main papers present material from four minicourses given by leading experts: D. Cartwright, A. Figà-Talamanca, S. Sawyer, and T. Steger. These minicrouses are introductions which gradually progress to deeper and less known branches of the subject. One of the topics treated is buildings, which are discrete analogues of symmetric spaces of arbitrary rank; buildings of rank are trees. Harmonic analysis on buildings is a fairly new and important field of research. One of the minicourses discusses buildings from the combinatorial perspective and another examines them from the p-adic perspective. the third minicourse deals with the connections of trees with p-adic analysis, and the fourth deals with random walks, ie., with the probabilistic side of harmonic functions on trees. The book also contains the extended abstracts of 19 of the 20 lectures given by the participants on their recent results. These abstracts, well detailed and clearly understandable, give a good cross-section of the present state of research in the field.
Author: Sheldon Axler Publisher: Springer Science & Business Media ISBN: 1475781377 Category : Mathematics Languages : en Pages : 266
Book Description
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: Library of Congress. Cataloging Policy and Support Office Publisher: ISBN: Category : Subject headings, Library of Congress Languages : en Pages : 1688
Author: F. Di Biase
Author: F. Di Biase
Author: J. L. Doob
Author: Jacob Korevaar
Author: Jordanka Paneva-konovska
Author: Peter Des Barres March
Author: Dorina Mitrea
Author: Library of Congress
Author: Adam Korǹyi (et al.)
Author: Sheldon Axler
Author: Library of Congress. Cataloging Policy and Support Office