The logarithmic asymptotic expansions for the norms of evaluation functionals

Dovgoshei A., ABDULLAYEV F., Kucukaslan M.

Siberian Mathematical Journal, vol.46, no.4, pp.613-622, 2005 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Review
  • Volume: 46 Issue: 4
  • Publication Date: 2005
  • Doi Number: 10.1007/s11202-005-0062-6
  • Journal Name: Siberian Mathematical Journal
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.613-622
  • Keywords: Evaluation functionals, General orthogonal polynomials, Green's function, Irregularity points for the Dirichlet problem, Logarithmic asymptotic expansion
  • Uşak University Affiliated: No

Abstract

Let μ be a compactly supported finite Borel measure in ℂ, and let Πn be the space of holomorphic polynomials of degree at most n furnished with the norm of L 2(μ). We study the logarithmic asymptotic expansions of the norms of the evaluation functionals that relate to polynomials p Πn their values at a point z ∈ ℂ. The main results demonstrate how the asymptotic behavior depends on regularity of the complement of the support of μ and the Stahl-Totik regularity of the measure. In particular, we study the cases of pointwise and μ-a.e. convergence as n → ∞. © 2005 Springer Science+Business Media, Inc.