Strong asymptotics for Sobolev orthogonal polynomials (original) (raw)

A new approach to the asymptotics for Sobolev orthogonal polynomials

arXiv (Cornell University), 2010

In this paper we deal with polynomials orthogonal with respect to an inner product involving derivatives, that is, a Sobolev inner product. Indeed, we consider Sobolev type polynomials which are orthogonal with respect to (f, g) = f gdµ + r i=0 M i f (i) (0)g (i) (0), M i ≥ 0, where µ is a certain probability measure with unbounded support. For these polynomials, we obtain the relative asymptotics with respect to orthogonal polynomials related to µ, Mehler-Heine type asymptotics and their consequences about the asymptotic behaviour of the zeros. To establish these results we use a new approach different from the methods used in the literature up to now. The development of this technique is highly motivated by the fact that the methods used when µ is bounded do not work.

A new approach to the asymptotics of Sobolev type orthogonal polynomials

Journal of Approximation Theory, 2011

This paper deals with Mehler-Heine type asymptotic formulas for the so-called discrete Sobolev orthogonal polynomials whose continuous part is given by Laguerre and generalized Hermite measures. We use a new approach which allows to solve the problem when the discrete part contains an arbitrary (finite) number of mass points. c

Asymptotics for varying discrete Sobolev orthogonal polynomials

Applied Mathematics and Computation, 2017

We consider a varying discrete Sobolev inner product such as (f, g) S = f (x) g(x) dμ + M n f (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 { M n } 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.

Sobolev orthogonal polynomials: Balance and asymptotics

Transactions of the American Mathematical Society, 2008

Let µ 0 and µ 1 be measures supported on an unbounded interval and S n,λn the extremal varying Sobolev polynomial which minimizes P, P λn = P 2 dµ 0 + λ n P 2 dµ 1 , λ n > 0 in the class of all monic polynomials of degree n. The goal of this paper is twofold. On one hand, we discuss how to balance both terms of this inner product, that is, how to choose a sequence (λ n) such that both measures µ 0 and µ 1 play a role in the asymptotics of (S n,λn). On the other, we apply such ideas to the case when both µ 0 and µ 1 are Freud weights. Asymptotics for the corresponding S n,λn are computed, illustrating the accuracy of the choice of λ n. * Partially supported by MEC of Spain under Grant MTM2006-13000-C03-03, FEDER funds (EU), and the DGA project E-64 (Spain). † Partially supported by MEC of Spain under Grant MTM2005-08648-C02-01 and Junta de Andalucía (FQM229 and excellence projects FQM481, P06-FQM-1735). ‡ Partially supported by MEC of Spain under Grants MTM 2004-03036 and MTM2006-13000-C03-03, FEDER funds (EU), and the DGA project E-64 (Spain).

Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

Acta Applicandae Mathematicae, 2006

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we focus on the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some directions for future research are formulated.