On the Method of Two-Sided Continuation of Solutions of the Integral Convolution Equation on a Finite Interval

The paper is devoted to the development of the method of two-sided continuation of the solution of the integral convolution equation
$$ S(x)=g(x)+\int _{0}^{r} K(x-t)S(t)\,dt,\qquad 0<x<r,\quad r< \infty, $$
with an even kernel function $K\in L_{1} (-r,r)$. Two continuations of the solution $S$ are considered: to $(-\infty, 0]$ and to $[r,\infty)$. A Wiener–Hopf-type factorization is used. Under invertibility conditions for some operators, the problem can be reduced to two equations with sum kernels:
$$ H^{\pm } (x)=q_{0}^{\pm } (x) \mp \int _{0}^{\infty } U(x+t+r)H^{\pm } (t)\,dt,\qquad x>0,\quad U\in L^{+} . $$
Applied aspects of the realization of the method are discussed.