On the Dirichlet problem for a~pseudodifferential equation encountered in the theory of random processes

B. V. Pal'tsev

Izvestiya. Mathematics , Tome 11 (1977) no. 6, pp. 1285-1322

Résumé

The problem is considered of finding a function $u(t)$ satisfying the equation \begin{equation} \mathscr F^{-1}[\tilde k(x)\tilde u(x)](t)=f(t)\quad\text{for}\quad t\in\Omega,\qquad\tilde u(x)=\mathscr F[u(t)](x), \end{equation} and the conditions \begin{equation} u(t)\equiv0\quad\text{for}\quad t\notin\Omega,\qquad\int_{-\infty}^{+\infty}\tilde k(x)|\tilde u(x)|^2\,dx\infty, \end{equation} where $\tilde k(x)$ is a nonnegative measurable function and $\mathscr F$ is the Fourier operator. An existence and uniqueness theorem is proved under quite general assumptions concerning the spectral densities $\tilde k(x)$. Explicit formulas for the solution of problem (1), (2) are obtained in the case when $\Omega$ is an interval $(-T,T)$ and $\tilde k(x)=|x|^\alpha$, $\alpha>0$. Bibliography: 17 titles.