General compactly supported solution of an integral equation of the convolution type

Abstract

We find the general form of solutions of the integral equation ∫k(t − s)u ₁(s) ds = u ₂(t) of the convolution type for the pair of unknown functions u ₁ and u ₂ in the class of compactly supported continuously differentiable functions under the condition that the kernel k(t) has the Fourier transform $\widetilde {{P_2}}$, where $\widetilde {{P_1}}$ and $\widetilde {{P_2}}$ are polynomials in the exponential e ^iτx, τ > 0, with coefficients polynomial in x. If the functions ${P_l}\left( x \right) = \widetilde {{P_l}}\left( {{e^{i\tau x}}} \right)$, l = 1, 2, have no common zeros, then the general solution in Fourier transforms has the form U _l (x) = P _l (x)R(x), l = 1, 2, where R(x) is the Fourier transform of an arbitrary compactly supported continuously differentiable function r(t).