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

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 $\tilde k(x)=|x|^\alpha$, $\alpha>0$.
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