Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights by Eli Levin & Doron S. Lubinsky

Author:Eli Levin & Doron S. Lubinsky
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

We shall show that

(9.31)

(9.32)

These together give (9.30). □

Proof of (9.31)

We assume that . We write for j 0 − K ≤ j ≤ j 0 + K − 1,

where

Here for some ξ ∈ I j , (9.21) gives

(9.33)

by (9.4), (9.5), (9.17), and (9.23). Then

(9.34)

for large enough n, if d n  ≥ C 0 for some fixed C 0. Then using the inequality (9.27) (which is applicable as )

(9.35)

Here

(9.36)

where Q is the conjugate Poisson kernel, given by

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