Some functional-difference equations solvable in finitary functions

The following equation is considered: $q(-i\partial/\partial x)u(x)=(f*u)(Ax)$, where $q$ is a polynomial with complex coefficients, $f$ is a compactly supported distribution, and $A\colon\mathbb{R}^n\to\mathbb{R}^n$ is a linear operator whose complexification has no spectrum in the closed unit disk. It turns out that this equation has a (smooth) solution $u(x)$ with compact support. In the one-dimensional case, this problem was treated earlier in detail by V. A. Rvachev and V. L. Rvachev and their numerous students.