Mathematical Notes

, Volume 97, Issue 3–4, pp 309–320

# On the method of two-sided continuation of solutions of the integral convolution equation on a finite Interval

• A. G. Barseghyan
Article

## Abstract

The paper is devoted to the development of the method of two-sided continuation of the solution of the integral convolution equation
$$\begin{array}{*{20}c} {S(x) = g(x) + \int_0^r {K(x - t)S(t)dt,} } & {0 < x < r,} & {r < \infty ,} \\ \end{array}$$
with an even kernel function KL 1(−r, r). Two continuations of the solution S are considered: to (−∞, 0] and to [r,∞). A Wiener–Hopf-type factorization is used. Under invertibility conditions for some operators, the problem can be reduced to two equations with sum kernels:
$$\begin{array}{*{20}c} {H^ \pm (x) = q_0^ \pm (x) \mp \int_0^\infty {U(x + t + r)H^ \pm (t)dt,} } & {x > 0,} & {U \in L^ + .} \\ \end{array}$$
Applied aspects of the realization of the method are discussed.

## Keywords

integral convolution equation two-sided continuation of a solution kernel function Wiener–Hopf-type factorization Baxter–Hirschman method

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