Asymptotics for varying discrete Sobolev orthogonal polynomials

摘要

We consider a varying discrete Sobolev inner product such as (f,g)S=∫f(x)g(x)dμ+Mnf(j)(c)g(j)(c),where μ is a finite positive Borel measure supported on an infinite subset of the real line, c is adequately located on the real axis, j≥0, and {Mn}n≥0 is a sequence of nonnegative real numbers satisfying a very general condition. Our aim is to study asymptotic properties of the sequence of orthonormal polynomials with respect to this Sobolev inner product. In this way, we focus our attention on Mehler-Heine type formulae as they describe in detail the asymptotic behavior of these polynomials around c, just the point where we have located the perturbation of the standard inner product. Moreover, we pay attention to the asymptotic behavior of the (scaled) zeros of these varying Sobolev polynomials and some numerical experiments are shown. Finally, we provide other asymptotic results which strengthen the idea that Mehler–Heine asymptotics describe in a precise way the differences between Sobolev orthogonal polynomials and standard ones.