The Fourier transform and convolutions generated by a differential operator with boundary condition on a segment

Abstract

We introduce the concepts of the Fourier transform and convolution generated by an arbitrary restriction of the differentiation operator in the space L2(0, b). In contrast to the classical convolution, the introduced convolution explicitly depends on the boundary condition that defines the domain of the operator L. The convolution is closely connected to the inverse operator or to the resolvent. So, we first find a representation for the resolvent, and then introduce the required convolution.