In this paper we introduce high dimensional tensor product interpolation for the combination technique. In order to compute the sparse grid solution, the discrete numerical subsolutions have to be extended by interpolation. If unsuitable interpolation techniques are used, the rate of convergence is deteriorated. We derive the necessary framework to preserve the error structure of high order finite difference solutions of elliptic partial differential equations within the combination technique framework. This strategy enables us to obtain high order sparse grid solutions on the full grid. As exemplifications for the case of order four we illustrate our theoretical results by two test examples with up to four dimensions.

Given a homogeneous elliptic partial differential operator $L$ with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain $\Omega$ in ${{\mathbf{R}}^{n}}$ and which belong locally to a Banach space $V$, we consider the problem of approximating in the norm of $V$ the functions in this class by “analytic” and “meromorphic” solutions of the equation $Lu\,=\,0$. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces $V$ and operators $L$. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.