Inversion of integral operators with kernels discontinuous on the diagonal

Abstract

Conditions implying the invertibility of the integral operator $$Af(x) = \int_0^1 {A(x,{\mathbf{ }}t)f(t){\mathbf{ }}dt}$$ with kernelA(x, t) having discontinuities of the first kind at the pointst=x andt=1−x are found. We give explicit inversion formulas as well as applications to the problem of finding the square roots of the operatory″(x) with arbitrary boundary conditions and the problem of expansion with respect to eigenfunctions.