Asymptotic Properties of Müntz Orthogonal Polynomials
Asymptotic Properties of Müntz Orthogonal Polynomials

Author: Úlfar F. Stefánsson

Publisher:

Published: 2010

Total Pages:

ISBN-13:

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Müntz 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.

Discrete Orthogonal Polynomials. (AM-164)
Discrete Orthogonal Polynomials. (AM-164)

Author: J. Baik

Publisher: Princeton University Press

Published: 2007-01-02

Total Pages: 179

ISBN-13: 1400837138

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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.

Asymptotics for Orthogonal Polynomials
Asymptotics for Orthogonal Polynomials

Author: Walter Van Assche

Publisher: Springer

Published: 2006-11-14

Total Pages: 207

ISBN-13: 354047711X

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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.

Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights
Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights

Author: Eli Levin

Publisher: Springer

Published: 2018-02-13

Total Pages: 168

ISBN-13: 3319729470

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This book establishes bounds and asymptotics under almost minimal conditions on the varying weights, and applies them to universality limits and entropy integrals. Orthogonal polynomials associated with varying weights play a key role in analyzing random matrices and other topics. This book will be of use to a wide community of mathematicians, physicists, and statisticians dealing with techniques of potential theory, orthogonal polynomials, approximation theory, as well as random matrices.

Asymptotic Properties of Polynomials with Auxiliary Conditions of Interpolation
Asymptotic Properties of Polynomials with Auxiliary Conditions of Interpolation

Author: Joseph Leonard Walsh

Publisher:

Published: 1961

Total Pages: 34

ISBN-13:

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Let a closed bounded point set E be given in the z-plane. Necessary and sufficient conditions are found for validity of the following: given assigned conditions of interpolation in a finite number of points: pn(zk) = Ank; there exists a sequence of polynomials pn(z) satisfying these conditions and lim sup max pn(z), z on E 1/n = (E), where (E) is the transfinite diameter of E. (Author).