A Class of Linear Integral Equations and Systems with Sum and Difference Kernel

Abstract

By means of Fourier transform and Cauchy integral techniques a complete investigation of a class of linear integral equations and corresponding systems of equations of cross-correlation type in the Lebesgue spaces L^1 and L^2 is performed. Integral equations of first and second kind are reduced to explicitly solvable Riemann-Hilbert problems for a holomorphic function in the upper half-plane and the system of equations to conjugacy problems for a sectionally holomorphic function, where in the case of a finite interval also the analytic continuation of the solutions to the lower half-plane can be carried out in explicit way. Further, a resolvent representation of the solution to the integral equation and its adjoint equation is derived.