A generalization of a singular convolution equation

Abstract

We consider one equation in the space of distributions which admit analytic representations (the Cauchy representations). This equation enables us to study an analog of the classic singular integrodifferential equation. It includes both the convolution equations and the ordinary linear differential equations, whose order is not greater than two, from the Fuchs class ([1], p. 202). It also contains linear differential equations with constant coefficients of any order in the mentioned space of distributions. Here the automorphism of the Fourier transform in the space of distributions of moderate growth allows us to solve equations which represent the Fourier images of the initial equations. Let Oα(R) stand for the space of functions which are infinitely differentiable on R and satisfy the condition φ(t) = O(|t|), |t| → ∞,