A~generalization of the Wiener--Hopf method for convolution equations on a~finite interval with symbols having power-like asymptotics at infinity

B. V. Pal'tsev

Sbornik. Mathematics, Tome 41 (1982) no. 3, pp. 289-328

Résumé

A generalization of the Wiener–Hopf method is obtained for convolution equations on the finite interval $(-T,T)$ $$ (\mathbf Ku)(t)=f(t),\qquad|t|, $$ where $\mathbf K$ is the convolution operator $\mathbf Ku(t)=(r_{(-T,T)}k*u)(t)$, $u(t)\in\mathscr S'(\mathbf R^1)$, $u(t)\equiv0$ for $|t|>T$, $*$ is the convolution operation, $k=k(t)$ is a kernel belonging to $\mathscr S'(\mathbf R^1)$, $r_{(-T,T)}$ is the operator of restriction of a generalized function to the interval $(-T,T)$, and $f(t)\in\mathscr D'(-T,T)$. Here $\mathscr S(\mathbf R^1)$ and $\mathscr S'(\mathbf R^1)$ are the Schwartz spaces of rapidly decreasing test functions and generalized functions of slow growth on $\mathbf R^1$, respectively. Bibliogrpahy: 19 titles.