Density results with linear combinations of translates of fundamental solutions

  • Authors:

  • Yiorgos-Sokratis Smyrlis

  • Affiliations:

  • Department of Mathematics and Statistics, University of Cyprus, Kallipoleos 75, P. O. Box 20537, 1678 Nicosia, Cyprus

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2009

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Abstract

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain @W by linear combinations of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie outside of @W@?. The domains under consideration may possess holes and they are required to satisfy a rather mild boundary regularity requirement, namely the segment condition. We study approximations with respect to the norms of the spaces C^k(@W@?) and the spaces of uniformly Holder continuous functions lip^k^,^@s(@W@?), and we establish density and non-density results for elliptic operators with constant coefficients. We also provide applications of our density results related to the method of fundamental solutions and to the theory of universal series.