TY - JOUR

T1 - Asymptotic behavior of orthogonal polynomials without the Carleman condition

AU - Yafaev, D.R.

N1 - Publisher Copyright: © 2020 Elsevier Inc.

PY - 2020/10/15

Y1 - 2020/10/15

N2 - Our goal is to find an asymptotic behavior as n→∞ of orthogonal polynomials P n(z) defined by the Jacobi recurrence coefficients a n,b n. We suppose that the off-diagonal coefficients a n grow so rapidly that the series ∑a n −1 converges, that is, the Carleman condition is violated. With respect to diagonal coefficients b n we assume that −b n(a na n−1) −1/2→2β ∞ for some β ∞≠±1. The asymptotic formulas obtained for P n(z) are quite different from the case ∑a n −1=∞ when the Carleman condition is satisfied. In particular, if ∑a n −1<∞, then the phase factors in these formulas do not depend on the spectral parameter z∈C. The asymptotic formulas obtained in the cases |β ∞|<1 and |β ∞|>1 are also qualitatively different from each other. As an application of these results, we find necessary and sufficient conditions for the essential self-adjointness of the corresponding minimal Jacobi operator.

AB - Our goal is to find an asymptotic behavior as n→∞ of orthogonal polynomials P n(z) defined by the Jacobi recurrence coefficients a n,b n. We suppose that the off-diagonal coefficients a n grow so rapidly that the series ∑a n −1 converges, that is, the Carleman condition is violated. With respect to diagonal coefficients b n we assume that −b n(a na n−1) −1/2→2β ∞ for some β ∞≠±1. The asymptotic formulas obtained for P n(z) are quite different from the case ∑a n −1=∞ when the Carleman condition is satisfied. In particular, if ∑a n −1<∞, then the phase factors in these formulas do not depend on the spectral parameter z∈C. The asymptotic formulas obtained in the cases |β ∞|<1 and |β ∞|>1 are also qualitatively different from each other. As an application of these results, we find necessary and sufficient conditions for the essential self-adjointness of the corresponding minimal Jacobi operator.

KW - Jacobi matrices

KW - Carleman condition

KW - orthogonal polynomials

KW - Asymptotics for large numbers

KW - Orthogonal polynomials

UR - https://www.sciencedirect.com/science/article/abs/pii/S0022123620301919#!

UR - http://www.scopus.com/inward/record.url?scp=85084789110&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2020.108648

DO - 10.1016/j.jfa.2020.108648

M3 - Article

VL - 279

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 7

M1 - 108648

ER -