Generalized Analytic Continuation
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| Author | : William T. Ross |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 165 |
| Release | : 2002 |
| Genre | : Mathematics |
| ISBN | : 0821831755 |
The theory of generalized analytic continuation studies continuations of meromorphic functions in situations where traditional theory says there is a natural boundary. This broader theory touches on a remarkable array of topics in classical analysis, as described in the book. The authors use the strong analogy with the summability of divergent series to motivate the subject. They are careful to cover the various types of continuations, attempting to unify them and suggesting some open questions. The book also addresses the role of such continuations in approximation theory and operator theory. The introductory overview provides a useful look at the history and context of the theory.
| Author | : I. N. Vekua |
| Publisher | : Elsevier |
| Total Pages | : 699 |
| Release | : 2014-07-17 |
| Genre | : Mathematics |
| ISBN | : 1483184676 |
Generalized Analytic Functions is concerned with foundations of the general theory of generalized analytic functions and some applications to problems of differential geometry and theory of shells. Some classes of functions and operators are discussed, along with the reduction of a positive differential quadratic form to the canonical form. Boundary value problems and infinitesimal bendings of surfaces are also considered. Comprised of six chapters, this volume begins with a detailed treatment of various problems of the general theory of generalized analytic functions as as well as boundary value problems. The reader is introduced to some classes of functions and functional spaces, with emphasis on functions of two independent variables. Subsequent chapters focus on the problem of reducing a positive differential quadratic form to the canonical form; basic properties of solutions of elliptic systems of partial differential equations of the first order, in a two-dimensional domain; and some boundary value problems for an elliptic system of equations of the first order and for an elliptic equation of the second order, in a two-dimensional domain. The final part of the book deals with problems of the theory of surfaces and the membrane theory of shells. This book is intended for students of advanced courses of the mechanico-mathematical faculties, postgraduates, and research workers.
| Author | : W.F.Jr. Donoghue |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 191 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642657559 |
A Pick function is a function that is analytic in the upper half-plane with positive imaginary part. In the first part of this book we try to give a readable account of this class of functions as well as one of the standard proofs of the spectral theorem based on properties of this class. In the remainder of the book we treat a closely related topic: Loewner's theory of monotone matrix functions and his analytic continuation theorem which guarantees that a real function on an interval of the real axis which is a monotone matrix function of arbitrarily high order is the restriction to that interval of a Pick function. In recent years this theorem has been complemented by the Loewner-FitzGerald theorem, giving necessary and sufficient conditions that the continuation provided by Loewner's theorem be univalent. In order that our presentation should be as complete and trans parent as possible, we have adjoined short chapters containing the in formation about reproducing kernels, almost positive matrices and certain classes of conformal mappings needed for our proofs.
| Author | : N K Bliev |
| Publisher | : CRC Press |
| Total Pages | : 160 |
| Release | : 1997-05-19 |
| Genre | : Mathematics |
| ISBN | : 9780582288614 |
This book studies the foundations of the general theory of generalized analytic functions in fractional spaces. The employment of fractional spaces and embedding theorems support applications of the theory of generalized analytic functions. The results obtained are applicable to the theory of singular integral equations, boundary value problems for elliptic differential equations, functions of a complex variable, as well as the theory of plates and shells. The book will be of interest to scientific workers and specialists interested in these questions and likewise to advanced students in mechanical engineering faculties.
| Author | : N.N. Bogolubov |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 714 |
| Release | : 2012-12-06 |
| Genre | : Science |
| ISBN | : 9400904916 |
The majority of the "memorable" results of relativistic quantum theory were obtained within the framework of the local quantum field approach. The explanation of the basic principles of the local theory and its mathematical structure has left its mark on all modern activity in this area. Originally, the axiomatic approach arose from attempts to give a mathematical meaning to the quantum field theory of strong interactions (of Yukawa type). The fields in such a theory are realized by operators in Hilbert space with a positive Poincare-invariant scalar product. This "classical" part of the axiomatic approach attained its modern form as far back as the sixties. * It has retained its importance even to this day, in spite of the fact that nowadays the main prospects for the description of the electro-weak and strong interactions are in connection with the theory of gauge fields. In fact, from the point of view of the quark model, the theory of strong interactions of Wightman type was obtained by restricting attention to just the "physical" local operators (such as hadronic fields consisting of ''fundamental'' quark fields) acting in a Hilbert space of physical states. In principle, there are enough such "physical" fields for a description of hadronic physics, although this means that one must reject the traditional local Lagrangian formalism. (The connection is restored in the approximation of low-energy "phe nomenological" Lagrangians.
| Author | : Alladi Ramakrishnan |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 203 |
| Release | : 2013-03-09 |
| Genre | : Science |
| ISBN | : 1468477218 |
This volume comprises the lectures given at the Fifth Anniversary Symposium held at the Institute of Mathematical Sciences, Madras, India, during January 1967. Professor Dallaporta of Padua delivered the inaugural address on the fundamental problem of quasars "whose study appears to hold im plications for cosmology." He presented a critical review of several attempts to understand their exceptionally large red shifts and also discussed the physical theories concerning the cause of the explosions which give rise to the quasars and to the tremendous energy output they require - questions which still remain unanswered. He stated, in concluding, that we may have to invoke certain aspects of the present theories of elementary particles in order to unravel these mysteries. Professor Mercier, well known for his studies on the philosophical foundations of modern physics, critically examined the various at tempts, such as that of Einstein, to formulate a unified field theory.
| Author | : J.F. Colombeau |
| Publisher | : Elsevier |
| Total Pages | : 297 |
| Release | : 2011-08-18 |
| Genre | : Mathematics |
| ISBN | : 0080872247 |
The author's previous book `New Generalized Functions and Multiplication of Distributions' (North-Holland, 1984) introduced `new generalized functions' in order to explain heuristic computations of Physics and to give a meaning to any finite product of distributions. The aim here is to present these functions in a more direct and elementary way. In Part I, the reader is assumed to be familiar only with the concepts of open and compact subsets of R&eegr;, of C∞ functions of several real variables and with some rudiments of integration theory. Part II defines tempered generalized functions, i.e. generalized functions which are, in some sense, increasing at infinity no faster than a polynomial (as well as all their partial derivatives). Part III shows that, in this setting, the partial differential equations have new solutions. The results obtained show that this setting is perfectly adapted to the study of nonlinear partial differential equations, and indicate some new perspectives in this field.
| Author | : I. M. Gel′fand |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 450 |
| Release | : 2016-04-19 |
| Genre | : Mathematics |
| ISBN | : 1470426587 |
he first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gel′fand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory. Volume 1 is devoted to basics of the theory of generalized functions. The first chapter contains main definitions and most important properties of generalized functions as functional on the space of smooth functions with compact support. The second chapter talks about the Fourier transform of generalized functions. In Chapter 3, definitions and properties of some important classes of generalized functions are discussed; in particular, generalized functions supported on submanifolds of lower dimension, generalized functions associated with quadratic forms, and homogeneous generalized functions are studied in detail. Many simple basic examples make this book an excellent place for a novice to get acquainted with the theory of generalized functions. A long appendix presents basics of generalized functions of complex variables.
| Author | : Bogdan Ziemian |
| Publisher | : |
| Total Pages | : 112 |
| Release | : 1996 |
| Genre | : Analytic functions |
| ISBN | : |
| Author | : Javad Mashreghi |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 330 |
| Release | : 2015-04-23 |
| Genre | : Mathematics |
| ISBN | : 1470410451 |
This volume contains the proceedings of the CRM Workshop on Invariant Subspaces of the Shift Operator, held August 26-30, 2013, at the Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada. The main theme of this volume is the invariant subspaces of the shift operator (or its adjoint) on certain function spaces, in particular, the Hardy space, Dirichlet space, and de Branges-Rovnyak spaces. These spaces, and the action of the shift operator on them, have turned out to be a precious tool in various questions in analysis such as function theory (Bieberbach conjecture, rigid functions, Schwarz-Pick inequalities), operator theory (invariant subspace problem, composition operator), and systems and control theory. Of particular interest is the Dirichlet space, which is one of the classical Hilbert spaces of holomorphic functions on the unit disk. From many points of view, the Dirichlet space is an interesting and challenging example of a function space. Though much is known about it, several important open problems remain, most notably the characterization of its zero sets and of its shift-invariant subspaces. This book is co-published with the Centre de Recherches Mathématiques.
