On orthogonal polynomials with respect to certain discrete Sobolev inner product

Abstract

In this paper we deal with sequences of polynomials orthogonal with respect to the discrete Sobolev inner productwhere ω is a weight function, ξ ≤ 0, and M, N ≥ 0. The location of the zeros of discrete Sobolev orthogonal polynomials is given in terms of the zeros of standard polynomials orthogonal with respect to the weight function ω. In particular, for ω(x) = x α e −x we obtain the asymptotics for discrete Laguerre-Sobolev orthogonal polynomials.

On Sobolev orthogonal polynomials

“…(1) (Outer relative asymptotics) lim The case that rank A = 1 is also studied, we only state the results when A has full rank for sake of simplicity. Finally, if the point c is a negative real number, then the following outer relative asymptotics was established in [70],…”

Connection formulas for general discrete Sobolev polynomials: Mehler–Heine asymptotics

“…Moreover, for k = j s , taking into account (13), (14) and the hypothesis for q n,s−1 , we can deduce…”

Asymptotic behavior of the partial derivatives of Laguerre kernels and some applications

“…, respectively. The following proposition summarizes some structural and asymptotic properties of the classical Laguerre polynomials (see [6,7,19] and the references therein).…”

“…In the setting of orthogonal polynomial theory these kernels have been especially used by Freud and Nevai [4,21,22] and, more recently, the remarkable Lubinsky's works [9,10] have caused heightened interest in this topic. Also, other interesting and related results corresponding to Fourier-Sobolev expansions may be found in [11][12][13][14][15][17][18][19]25].…”