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Author : J. Baik,T. Kriecherbauer,Kenneth D.T-R McLaughlin,Peter D. Miller
Publisher : Princeton University Press
Page : 178 pages
File Size : 44,6 Mb
Release : 2007
Category : Mathematics
ISBN : 9780691127347
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Author : J. Baik,T. Kriecherbauer,Kenneth D.T-R McLaughlin,Peter D. Miller
Publisher : Princeton University Press
Page : 179 pages
File Size : 47,8 Mb
Release : 2007-01-02
Category : Mathematics
ISBN : 9781400837137
This book describes the theory and applications of discrete orthogonal polynomials--polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.
Author : Arnold F. Nikiforov,Sergei K. Suslov,Vasilii B. Uvarov
Publisher : Springer Science & Business Media
Page : 388 pages
File Size : 48,8 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783642747489
While classical orthogonal polynomials appear as solutions to hypergeometric differential equations, those of a discrete variable emerge as solutions of difference equations of hypergeometric type on lattices. The authors present a concise introduction to this theory, presenting at the same time methods of solving a large class of difference equations. They apply the theory to various problems in scientific computing, probability, queuing theory, coding and information compression. The book is an expanded and revised version of the first edition, published in Russian (Nauka 1985). Students and scientists will find a useful textbook in numerical analysis.
Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 40,8 Mb
Release : 2007
Category : Electronic
ISBN : OCLC:909902839
Author : Mourad Ismail
Publisher : Cambridge University Press
Page : 748 pages
File Size : 47,6 Mb
Release : 2005-11-21
Category : Mathematics
ISBN : 0521782015
The first modern treatment of orthogonal polynomials from the viewpoint of special functions is now available in paperback.
Author : Francisco Marcellàn
Publisher : Springer Science & Business Media
Page : 432 pages
File Size : 49,8 Mb
Release : 2006-06-19
Category : Mathematics
ISBN : 9783540310624
Special functions and orthogonal polynomials in particular have been around for centuries. Can you imagine mathematics without trigonometric functions, the exponential function or polynomials? In the twentieth century the emphasis was on special functions satisfying linear differential equations, but this has now been extended to difference equations, partial differential equations and non-linear differential equations. The present set of lecture notes containes seven chapters about the current state of orthogonal polynomials and special functions and gives a view on open problems and future directions. The topics are: computational methods and software for quadrature and approximation, equilibrium problems in logarithmic potential theory, discrete orthogonal polynomials and convergence of Krylov subspace methods in numerical linear algebra, orthogonal rational functions and matrix orthogonal rational functions, orthogonal polynomials in several variables (Jack polynomials) and separation of variables, a classification of finite families of orthogonal polynomials in Askey’s scheme using Leonard pairs, and non-linear special functions associated with the Painlevé equations.
Author : Mourad E. H. Ismail,Walter Van Assche
Publisher : Cambridge University Press
Page : 0 pages
File Size : 50,9 Mb
Release : 2020-09-17
Category : Mathematics
ISBN : 9780521197427
Extensive update of the Bateman Manuscript Project. Volume 1 covers orthogonal polynomials and moment problems.
Author : Mama Foupouagnigni,Wolfram Koepf
Publisher : Springer Nature
Page : 683 pages
File Size : 44,7 Mb
Release : 2020-03-11
Category : Mathematics
ISBN : 9783030367442
This book presents contributions of international and local experts from the African Institute for Mathematical Sciences (AIMS-Cameroon) and also from other local universities in the domain of orthogonal polynomials and applications. The topics addressed range from univariate to multivariate orthogonal polynomials, from multiple orthogonal polynomials and random matrices to orthogonal polynomials and Painlevé equations. The contributions are based on lectures given at the AIMS-Volkswagen Stiftung Workshop on Introduction of Orthogonal Polynomials and Applications held on October 5–12, 2018 in Douala, Cameroon. This workshop, funded within the framework of the Volkswagen Foundation Initiative "Symposia and Summer Schools", was aimed globally at promoting capacity building in terms of research and training in orthogonal polynomials and applications, discussions and development of new ideas as well as development and enhancement of networking including south-south cooperation.
Author : Edward B. Saff,Vilmos Totik
Publisher : Springer Science & Business Media
Page : 517 pages
File Size : 55,7 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9783662033296
In recent years approximation theory and the theory of orthogonal polynomials have witnessed a dramatic increase in the number of solutions of difficult and previously untouchable problems. This is due to the interaction of approximation theoretical techniques with classical potential theory (more precisely, the theory of logarithmic potentials, which is directly related to polynomials and to problems in the plane or on the real line). Most of the applications are based on an exten sion of classical logarithmic potential theory to the case when there is a weight (external field) present. The list of recent developments is quite impressive and includes: creation of the theory of non-classical orthogonal polynomials with re spect to exponential weights; the theory of orthogonal polynomials with respect to general measures with compact support; the theory of incomplete polynomials and their widespread generalizations, and the theory of multipoint Pade approximation. The new approach has produced long sought solutions for many problems; most notably, the Freud problems on the asymptotics of orthogonal polynomials with a respect to weights of the form exp(-Ixl ); the "l/9-th" conjecture on rational approximation of exp(x); and the problem of the exact asymptotic constant in the rational approximation of Ixl. One aim of the present book is to provide a self-contained introduction to the aforementioned "weighted" potential theory as well as to its numerous applications. As a side-product we shall also fully develop the classical theory of logarithmic potentials.
Author : Walter Van Assche
Publisher : Cambridge University Press
Page : 192 pages
File Size : 41,8 Mb
Release : 2018
Category : Mathematics
ISBN : 9781108441940
There are a number of intriguing connections between Painlev equations and orthogonal polynomials, and this book is one of the first to provide an introduction to these. Researchers in integrable systems and non-linear equations will find the many explicit examples where Painlev equations appear in mathematical analysis very useful. Those interested in the asymptotic behavior of orthogonal polynomials will also find the description of Painlev transcendants and their use for local analysis near certain critical points helpful to their work. Rational solutions and special function solutions of Painlev equations are worked out in detail, with a survey of recent results and an outline of their close relationship with orthogonal polynomials. Exercises throughout the book help the reader to get to grips with the material. The author is a leading authority on orthogonal polynomials, giving this work a unique perspective on Painlev equations.
Author : Gabor Szeg
Publisher : American Mathematical Soc.
Page : 448 pages
File Size : 52,9 Mb
Release : 1939-12-31
Category : Mathematics
ISBN : 9780821810231
The general theory of orthogonal polynomials was developed in the late 19th century from a study of continued fractions by P. L. Chebyshev, even though special cases were introduced earlier by Legendre, Hermite, Jacobi, Laguerre, and Chebyshev himself. It was further developed by A. A. Markov, T. J. Stieltjes, and many other mathematicians. The book by Szego, originally published in 1939, is the first monograph devoted to the theory of orthogonal polynomials and its applications in many areas, including analysis, differential equations, probability and mathematical physics. Even after all the years that have passed since the book first appeared, and with many other books on the subject published since then, this classic monograph by Szego remains an indispensable resource both as a textbook and as a reference book. It can be recommended to anyone who wants to be acquainted with this central topic of mathematical analysis.
Author : Howard S. Cohl,Mourad E. H. Ismail
Publisher : Cambridge University Press
Page : 351 pages
File Size : 44,5 Mb
Release : 2020-10-15
Category : Mathematics
ISBN : 9781108821599
Contains graduate-level introductions by international experts to five areas of research in orthogonal polynomials and special functions.
Author : Walter Van Assche
Publisher : Springer
Page : 207 pages
File Size : 52,5 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540477112
Recently there has been a great deal of interest in the theory of orthogonal polynomials. The number of books treating the subject, however, is limited. This monograph brings together some results involving the asymptotic behaviour of orthogonal polynomials when the degree tends to infinity, assuming only a basic knowledge of real and complex analysis. An extensive treatment, starting with special knowledge of the orthogonality measure, is given for orthogonal polynomials on a compact set and on an unbounded set. Another possible approach is to start from properties of the coefficients in the three-term recurrence relation for orthogonal polynomials. This is done using the methods of (discrete) scattering theory. A new method, based on limit theorems in probability theory, to obtain asymptotic formulas for some polynomials is also given. Various consequences of all the results are described and applications are given ranging from random matrices and birth-death processes to discrete Schrödinger operators, illustrating the close interaction with different branches of applied mathematics.
Author : Roelof Koekoek,Peter A. Lesky,René F. Swarttouw
Publisher : Springer
Page : 578 pages
File Size : 50,9 Mb
Release : 2010-10-22
Category : Mathematics
ISBN : 3642050506
The present book is about the Askey scheme and the q-Askey scheme, which are graphically displayed right before chapter 9 and chapter 14, respectively. The fa- lies of orthogonal polynomials in these two schemes generalize the classical orth- onal polynomials (Jacobi, Laguerre and Hermite polynomials) and they have pr- erties similar to them. In fact, they have properties so similar that I am inclined (f- lowing Andrews & Askey [34]) to call all families in the (q-)Askey scheme classical orthogonal polynomials, and to call the Jacobi, Laguerre and Hermite polynomials very classical orthogonal polynomials. These very classical orthogonal polynomials are good friends of mine since - most the beginning of my mathematical career. When I was a fresh PhD student at the Mathematical Centre (now CWI) in Amsterdam, Dick Askey spent a sabbatical there during the academic year 1969–1970. He lectured to us in a very stimulating wayabouthypergeometricfunctionsandclassicalorthogonalpolynomials. Evenb- ter, he gave us problems to solve which might be worth a PhD. He also pointed out to us that there was more than just Jacobi, Laguerre and Hermite polynomials, for instance Hahn polynomials, and that it was one of the merits of the Higher Transc- dental Functions (Bateman project) that it included some newer stuff like the Hahn polynomials (see [198, §10. 23]).
Author : Walter Gautschi
Publisher : SIAM
Page : 335 pages
File Size : 43,5 Mb
Release : 2016-05-23
Category : Science
ISBN : 9781611974300
Techniques for generating orthogonal polynomials numerically have appeared only recently, within the last 30 or so years.?Orthogonal Polynomials in MATLAB: Exercises and Solutions?describes these techniques and related applications, all supported by MATLAB programs, and presents them in a unique format of exercises and solutions designed by the author to stimulate participation. Important computational problems in the physical sciences are included as models for readers to solve their own problems.?
