Asymptotics of Sobolev Orthogonal Polynomials for Coherent Pairs of Laguerre Type

Abstract

Let S n n denote a sequence of polynomials orthogonal with respect to the Sobolev inner product f g S = f x g x dψ 0 x + λ f x g x dψ 1 x where λ > 0 and dψ 0 dψ 1 is a so-called coherent pair with at least one of the measures dψ 0 or dψ 1 a Laguerre measure. We investigate the asymptotic behaviour of S n x outside the supports of dψ 0 and dψ 1 , and n → +∞.

Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

“…Indeed, this situation is the only one studied for measures with unbounded supports until the work [10] for an example of Freud weights. Three contributions concerning Laguerre and Hermite weights were given in the 90's (see [27], [41], [37]). …”

“…n (x)/T n (x) (see [37,Remark 4.6]). On the other hand, the zeros of R (L2,∞) n (x) accumulate on [0, ∞) when n → ∞.…”

A Cohen-Type Inequality for Fourier Expansions with Respect to Non-Discrete Laguerre–Sobolev Inner Product

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“…[2,11]. We have the following relation between Laguerre polynomials and Laguerre-Sobolev polynomials…”

Companion Linear Functionals and Sobolev Inner Products: A Case Study

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