Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

Abstract

9 pages, no figures.-- MSC2000 code: 33C45.MR#: MR2431543 (2009g:41009)Zbl#: Zbl 1155.33006We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same $n$th root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e−φ(x), giving a unified treatment for the so-called Freud (i.e., when φ has polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.Research by first two authors (C.D.M. and R.O.) was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grants MTM2005-08571 and MTM2007-68114. Research by third author (H.P.) was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grant MTM2006-13000-C03-02, by Comunidad de Madrid-Universidad Carlos III de Madrid, under grants CCG06-UC3M/EST-0690 and CCG07-UC3M/ESP-3339, by Centro de Investigación Matemática de Canarias (CIMAC) and by Vicerrectorado de Investigación de La Universidad de La Laguna: Convocatoria 2005 de Ayudas a Profesores Invitados.Publicad