Universal approximation by translates of fundamental solutions of elliptic equations

Abstract

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain ω by universal series of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie on a prescribed surface outside of, known as the pseudo-boundary. The domains under consideration satisfy a rather mild boundary regularity requirement, namely, the segment condition. We study approximations with respect to the norms of the spaces and we establish the existence of universal series. Analogous results are obtainable with respect to the norms of Holder spaces The sequence of coefficients of the universal series may be chosen in but it can not be chosen in. © 2011, by Oldenbourg Wissenschaftsverlag, Nicosia, Germany. All rights reserved.