A new approach to the convolution operator on a finite interval

The convolution operator on a finite interval defined on a space ofL ₂ functions is studied by relating it to a singular integral operator acting on a space of functions defined on a system of two parallel straight lines in the complex plane . The approach followed in the paper applies both to the case where the Fourier transform of the kernel functions is anL function and to the case where the kernel function is periodic, thus yielding a unified treatment of these two classes of kernel functions. In the non-periodic case it is possible, for a special class of kernel functions, to study the invertibility property of the operator giving an explicit formula for the inverse. An example is presented and generalizations are suggested.