Some Asymptotic Properties of Orthogonal Polynomials
Author: Maria P. Alfaro
Publisher:
Published: 1992
Total Pages: 14
ISBN-13:
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Author: Maria P. Alfaro
Publisher:
Published: 1992
Total Pages: 14
ISBN-13:
DOWNLOAD EBOOKAuthor: Walter Van Assche
Publisher: Springer
Published: 2006-11-14
Total Pages: 207
ISBN-13: 354047711X
DOWNLOAD EBOOKRecently 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: J. Baik
Publisher: Princeton University Press
Published: 2007-01-02
Total Pages: 179
ISBN-13: 1400837138
DOWNLOAD EBOOKThis 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: Walter Van Assche
Publisher:
Published: 1984
Total Pages: 26
ISBN-13:
DOWNLOAD EBOOKAuthor: Úlfar F. Stefánsson
Publisher:
Published: 2010
Total Pages:
ISBN-13:
DOWNLOAD EBOOKMüntz polynomials arise from consideration of Müntz's Theorem, which is a beautiful generalization of Weierstrass's Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials on the interval of orthogonality, and in particular obtain new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics and endpoint limit asymptotics on the interval. The zero spacing behavior follows, as well as estimates for the smallest and largest zeros. This is the first time that such asymptotics have been obtained for general Müntz exponents. We also look at the asymptotic behavior outside the interval and the asymptotic properties of the associated Christoffel functions.
Author: Gabor Szeg
Publisher: American Mathematical Soc.
Published: 1939-12-31
Total Pages: 448
ISBN-13: 0821810235
DOWNLOAD EBOOKThe 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: I͡A. L. Geronimus
Publisher:
Published: 1953
Total Pages: 36
ISBN-13:
DOWNLOAD EBOOKAuthor: Brian Zachary Simanek
Publisher:
Published: 2012
Total Pages: 0
ISBN-13:
DOWNLOAD EBOOKAuthor: Jakov L. Geronimus
Publisher:
Published: 1953
Total Pages: 29
ISBN-13:
DOWNLOAD EBOOKAuthor: L︠A︡. L. Geronimus
Publisher:
Published: 1953
Total Pages: 29
ISBN-13:
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