Sobolev orthogonal polynomials: Balance and asymptotics

Abstract

Abstract. Let µ 0 and µ 1 be measures supported on an unbounded interval and S n,λ n the extremal varying Sobolev polynomial which minimizesin the class of all monic polynomials of degree n. The goal of this paper is twofold. On the 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,λ n . On the other hand, we apply such ideas to the case when both µ 0 and µ 1 are Freud weights. Asymptotics for the corresponding S n,λ n are computed, illustrating the accuracy of the choice of λ n .

Asymptotics for Jacobi–Sobolev orthogonal polynomials associated with non-coherent pairs of measures

“…We are going to explain this affirmation. Of course, the ideas exposed here are different from the method developed in [1] and later in [2], but we can establish a relation between Theorem 3.3 and Theorem 1 in [1].…”

“…However, we point out that the techniques developed to balance Sobolev inner products in [1,2] are more general.…”

“…Furthermore, ψ 1 has a mass point outside (−1, 1). Outer strong asymptotics for general Sobolev orthogonal polynomials have been obtained by Martínez-Finkelshtein in [10] when the measures are absolutely continuous and supported on a smooth Jordan closed curve; • from the point of view of asymptotics properties the measure in (2) playing the most important role is ψ 1 because we have a quadratic factor n 2 out of the derivatives when we work with monic polynomials (see the surveys [9,11]). Here, this situation is also true but we will show that the inner product (2) is equilibrated in some sense.…”

“…Our objective is to study asymptotic properties of the polynomials orthogonal with respect to the inner product given in (2). However, it is very important to emphasize two points:…”

“…Here, this situation is also true but we will show that the inner product (2) is equilibrated in some sense. Notice that the technique used here is different from the one used in [1,2] where varying Sobolev orthogonal polynomials are considered.…”