A lacunary version of Mergelian's approximation theorem

  • Authors:

  • T. Gharibyan;W. Luh;J. Müller

  • Affiliations:

  • Institute of Mathematics, Armenian National Academy of Sciences, 375019 Yerevan, Armenia;University of Trier, FB IV, Mathematics, D-54286 Trier, Germany;University of Trier, FB IV, Mathematics, D-54286 Trier, Germany

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2010

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Abstract

Let K be a compact plane set having connected complement. Then Mergelian's theorem states that the linear span of the monomials z^n, i.e. the polynomials, are dense in the Banach space A(K) of all functions continuous on K and holomorphic in the interior of K endowed with the sup-norm. We consider the question under which conditions the linear span of z^n, with n running through a sequence of nonnegative integers having upper density one, is dense in A(K) or appropriate subspaces.