Asymptotic behavior of orthogonal polynomials. Singular critical case

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Asymptotic behavior of orthogonal polynomials. Singular critical case. / Yafaev, D.R. .

в: Journal of Approximation Theory, Том 262, 105506, 02.2021.

Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование

BibTeX

@article{745baea7e30b44b5b430b1e8d56edd61,

title = "Asymptotic behavior of orthogonal polynomials. Singular critical case",

abstract = "Our goal is to find an asymptotic behavior as n→∞ of the orthogonal polynomials P n(z) defined by Jacobi recurrence coefficients a n (off-diagonal terms) and b n (diagonal terms). We consider the case a n→∞, b n→∞ in such a way that ∑a n −1<∞ (that is, the Carleman condition is violated) and γ n:=2 −1b n(a na n−1) −1∕2→γ as n→∞. In the case |γ|≠1 asymptotic formulas for P n(z) are known; they depend crucially on the sign of |γ|−1. We study the critical case |γ|=1. The formulas obtained are qualitatively different in the cases |γ n|→1−0 and |γ n|→1+0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of P n(z) based on a close analogy of the Jacobi difference equations and differential equations of Schr{\"o}dinger type. ",

keywords = "Increasing Jacobi coefficients, Carleman condition, Difference equations, Jost solutions",

author = "D.R. Yafaev",

note = "Publisher Copyright: {\textcopyright} 2020",

year = "2021",

month = feb,

doi = "10.1016/j.jat.2020.105506",

language = "English",

volume = "262",

journal = "Journal of Approximation Theory",

issn = "0021-9045",

publisher = "Elsevier",

}

RIS

TY - JOUR

T1 - Asymptotic behavior of orthogonal polynomials. Singular critical case

AU - Yafaev, D.R.

PY - 2021/2

Y1 - 2021/2

N2 - Our goal is to find an asymptotic behavior as n→∞ of the orthogonal polynomials P n(z) defined by Jacobi recurrence coefficients a n (off-diagonal terms) and b n (diagonal terms). We consider the case a n→∞, b n→∞ in such a way that ∑a n −1<∞ (that is, the Carleman condition is violated) and γ n:=2 −1b n(a na n−1) −1∕2→γ as n→∞. In the case |γ|≠1 asymptotic formulas for P n(z) are known; they depend crucially on the sign of |γ|−1. We study the critical case |γ|=1. The formulas obtained are qualitatively different in the cases |γ n|→1−0 and |γ n|→1+0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of P n(z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type.

AB - Our goal is to find an asymptotic behavior as n→∞ of the orthogonal polynomials P n(z) defined by Jacobi recurrence coefficients a n (off-diagonal terms) and b n (diagonal terms). We consider the case a n→∞, b n→∞ in such a way that ∑a n −1<∞ (that is, the Carleman condition is violated) and γ n:=2 −1b n(a na n−1) −1∕2→γ as n→∞. In the case |γ|≠1 asymptotic formulas for P n(z) are known; they depend crucially on the sign of |γ|−1. We study the critical case |γ|=1. The formulas obtained are qualitatively different in the cases |γ n|→1−0 and |γ n|→1+0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of P n(z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type.

KW - Increasing Jacobi coefficients

KW - Carleman condition

KW - Difference equations

KW - Jost solutions

UR - https://www.sciencedirect.com/science/article/abs/pii/S0021904520301428#!

UR - http://www.scopus.com/inward/record.url?scp=85095679362&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2020.105506

DO - 10.1016/j.jat.2020.105506

M3 - Article

VL - 262

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

M1 - 105506

ER -

ID: 71379289

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