A simple characterization of weighted Sobolev spaces with bounded multiplication operator

  • Authors:

  • Jose M. Rodriguez

  • Affiliations:

  • Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911 Leganés (Madrid), Spain

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2008

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Abstract

In this paper we give a simple characterization of weighted Sobolev spaces (with piecewise monotone weights) such that the multiplication operator is bounded: it is bounded if and only if the support of @m"0 is large enough. We also prove some basic properties of the appropriate weighted Sobolev spaces. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials.