Sobolev Spaces with Respect to Measures in Curves and Zeros of Sobolev Orthogonal Polynomials Articles uri icon

publication date

  • December 2008

start page

  • 325

end page

  • 353

issue

  • 3

volume

  • 104

international standard serial number (ISSN)

  • 0167-8019

electronic international standard serial number (EISSN)

  • 1572-9036

abstract

  • In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. 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. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces.