The convolution equation of the first kind on a finite interval \[ \int^{1}_{0}k(x-\xi)\phi (\xi)d\xi =f(x),\quad x\in (0,1) \] is studied under assumption that k(x) is a matrix-valued function, f and \(\phi\) are vector-valued functions. The aim of the investigation is to obtain Fredholm properties of this equation - in the usual setting when \(\phi\) and f belong to the different Sobolev-type space connected with “restriction” of functions in \(L^{\mu}_ 2({\mathbb{R}}^ 1)\) onto [0,1].
The authors give some theorems on the Fredholm properties and the invertibiliy of the Wiener-Hopf operator associated with the above equation. They also study the example arising from wave diffraction by a strip.