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

TL;DR

This paper develops a new Fourier transform and convolution based on a differential operator with boundary conditions, linking these concepts to the operator's resolvent in the space L2(0,b).

Contribution

It introduces a boundary-condition-dependent convolution generated by a differential operator, extending classical Fourier analysis to more general boundary-restricted operators.

Findings

Derived a representation for the resolvent of the differential operator.

Defined a new convolution explicitly depending on boundary conditions.

Connected the convolution to the inverse operator and resolvent.

Abstract

We introduce the concepts of the Fourier transform and convolution generated by an arbitrary restriction of the differentiation operator in the space $L_{2}(0,b).$ 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.