An equation of convolution type on convex domains in~$\mathbf R^2$

V. V. Napalkov

Sbornik. Mathematics, Tome 23 (1974) no. 2, pp. 169-184

Résumé

Let $D$ be a convex domain in $\mathbf R^2$, and let $C^{(k)}(D)$, $k=(k_1,\,k_2)$, be the space of functions $f(x)$, continuous in $D$ together with their partial derivatives $$ \frac{\partial^{n_1+n_2}}{\partial x_1^{n_1}\partial x_2^{n_2}}f, $$ $n_1\leqslant k_1$, $n_2\leqslant k_2$. This space is provided with the natural topology of uniform convergence of functions and corresponding derivatives on compact subsets of $D$. Consider in $C^{(k)}(D)$ the homogeneous convolution equation $\mu*f=0$, where $\mu$ is a continuous linear functional on $C^{(k)}(D)$. It is proved that every solution of this equation from the space $C^{(k)}(D)$ can be approximated in the topology of $C^{(k)}(D)$ by a linear combination of exponential polynomials satisfying this equation. Bibliography: 15 titles.