Some asymptotics for Sobolev orthogonal polynomials involving Gegenbauer weights (original) (raw)

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 of Orthogonal Polynomials for a Weight with a Jump on [−1,1

Constructive Approximation, 2011

We consider the orthogonal polynomials on [−1,1] with respect to the weight w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \varXi _{c}(x),\quad\alpha,\beta>-1,$$ where h is real analytic and strictly positive on [−1,1] and Ξ c is a step-like function: Ξ c (x)=1 for x∈[−1,0) and Ξ c (x)=c 2, c>0, for x∈[0,1]. We obtain strong uniform asymptotics of the monic orthogonal polynomials in ℂ, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as n→∞. In particular, we prove for w c a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel–Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit clock behavior. For the asymptotic analysis we use the steepest descent method of Deift and Zhou applied to the noncommutative Riemann–Hilbert problems characterizing the orthogonal polynomials. The local analysis at x=0 is carried out in terms of confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.

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.