Finite interval convolution operators with transmission property (Q814347)

The present paper deals with convolution equations \[ K_a \phi (x)\equiv \phi (x)+\int_0^a k_a (x-y) \phi (y) dy =f(x) \] on a finite real interval in the framework of Bessel potential spaces and Sobolev spaces. The main focus is the corresponding transmission property. In the case of invertibility, the inverse of \(K_a\) is presented in terms of the canonical factorization of the related Fourier symbol matrix function.