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

For a homogeneous elliptic partial differential operator $L$ with constant coefficients and a class of functions (jet-distributions) defined on a closed, not necessarily compact, subset of $\mathbb R^n$ and belonging locally to a Banach space $V$, the approximation in the norm of $V$ of functions in this class by entire and meromorphic solutions of the equation $Lu=0$ is considered. Theorems of Runge, Mergelyan, Roth, and Arakelyan type are established for a wide class of Banach spaces $V$ and operators $L$ they encompass most of the previously considered generalizations of these theorems but also include new results.