A new approach to the asymptotics for Sobolev orthogonal polynomials

2010/03/17 by M. Alfaro, Manuel Alfaro, Juan J. Moreno–Balcázar +9
Mathematics · #33C45 #42C05 #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #math.CA #msc:33C45 #msc:42C05

paper · pdf · doi:10.48550/arxiv.1003.3336

31 pages

arxiv created 2010/03/17 · openalex publication_date 2010/03/17 · arxiv updated 2010/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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)=∫ fg dμ+∑_i=0^r 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.

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