Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions

Abstract

For a homogeneous elliptic partial differential operator with constant coefficients and a class of functions (jet-distributions) defined on a closed, not necessarily compact, subset of and belonging locally to a Banach space , the approximation in the norm of of functions in this class by entire and meromorphic solutions of the equation is considered. Theorems of Runge, Mergelyan, Roth, and Arakelyan type are established for a wide class of Banach spaces and operators they encompass most of the previously considered generalizations of these theorems but also include new results.