Joint Approximations of Distributions in Banach Spaces

For a given homogeneous elliptic partial differential operator $L$ with constant complex coefficients, two Banach spaces $V_1$ and $V_2$ of distributions in $\mathbb R^N$, and compact sets $X_1$ and $X_2$ in $\mathbb R^N$, we study joint approximations in the norms of the spaces $V_1(X_1)$ and $V_2(X_2)$ (the spaces of Whitney jet-distributions) by the solutions of the equation $L_u=0$ in neighborhoods of the set $X_1\cup X_2$. We obtain a localization theorem, which, under certain conditions, allows one to reduce the above-cited approximation problem to the corresponding separate problems in each of the spaces.