Uniform asymptotic formula for orthogonal polynomials with exponential weight

W. Y. Qiu; R. Wong

doi:10.1137/S0036141098344671

Uniform asymptotic formula for orthogonal polynomials with exponential weight

W. Y. Qiu
, R. Wong

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

12 Citations (Scopus)

22 Downloads (CityUHK Scholars)

Abstract

Let {p_n(x)}_n≥0 be the set of orthonormal polynomials with respect to the exponential weight w(x) = e^-v(x), where v(x) = x^2m + ⋯ is a monic polynomial of degree 2m with m ≥ 2 and is even. An asymptotic approximation is obtained for p_n(x), as n → ∞, which holds uniformly for 0 ≤ x ≤ O(n^1/2m. As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.

Original language	English
Pages (from-to)	992-1029
Journal	SIAM Journal on Mathematical Analysis
Volume	31
Issue number	5
DOIs	https://doi.org/10.1137/S0036141098344671
Publication status	Published - Apr 2000

Research Keywords

Exponential weight
Orthogonal polynomials
Turning point
Uniform asymptotic approximation
Zeros

Publisher's Copyright Statement

COPYRIGHT TERMS OF DEPOSITED FINAL PUBLISHED VERSION FILE: © 2000 Society for Industrial and Applied Mathematics.

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title = "Uniform asymptotic formula for orthogonal polynomials with exponential weight",

abstract = "Let \{pn(x)\}n≥0 be the set of orthonormal polynomials with respect to the exponential weight w(x) = e-v(x), where v(x) = x2m + ⋯ is a monic polynomial of degree 2m with m ≥ 2 and is even. An asymptotic approximation is obtained for pn(x), as n → ∞, which holds uniformly for 0 ≤ x ≤ O(n1/2m. As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.",

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TY - JOUR

T1 - Uniform asymptotic formula for orthogonal polynomials with exponential weight

AU - Qiu, W. Y.

AU - Wong, R.

PY - 2000/4

Y1 - 2000/4

N2 - Let {pn(x)}n≥0 be the set of orthonormal polynomials with respect to the exponential weight w(x) = e-v(x), where v(x) = x2m + ⋯ is a monic polynomial of degree 2m with m ≥ 2 and is even. An asymptotic approximation is obtained for pn(x), as n → ∞, which holds uniformly for 0 ≤ x ≤ O(n1/2m. As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.

AB - Let {pn(x)}n≥0 be the set of orthonormal polynomials with respect to the exponential weight w(x) = e-v(x), where v(x) = x2m + ⋯ is a monic polynomial of degree 2m with m ≥ 2 and is even. An asymptotic approximation is obtained for pn(x), as n → ∞, which holds uniformly for 0 ≤ x ≤ O(n1/2m. As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.

KW - Exponential weight

KW - Orthogonal polynomials

KW - Turning point

KW - Uniform asymptotic approximation

KW - Zeros

UR - https://www.scopus.com/pages/publications/0034345311

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-0034345311&origin=recordpage

U2 - 10.1137/S0036141098344671

DO - 10.1137/S0036141098344671

M3 - RGC 21 - Publication in refereed journal

SN - 0036-1410

VL - 31

SP - 992

EP - 1029

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 5

ER -

Uniform asymptotic formula for orthogonal polynomials with exponential weight

Abstract

Research Keywords

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