Operational Properties of the Hartley Convolution and Its Applications

In this paper, we study the fundamental properties of the Hartley convolution operator and its applications, which specifically are the normed ring structures on the L 1 ( R ) space when being equipped with the multiplication defined as Hartley convolution multiplication. We prove the Young-type theorem and Young’s inequalities for the Hartley convolution operator in the one-dimensional case. Furthermore, we also formulate the Watson-type transformation for the Hartley convolution operator. We establish necessary and sufficient conditions for this operator to be unitary on L 2 ( R ) and get its inverse represented in the conjugate symmetric form. In addition to showing properties of this operator, we use the obtained results to study the solvability in close-form and estimate the boundedness solutions of some classes for integro-differential equations of Barbashin type and the Fredholm-type integral equation. Finally, we build numerical examples for illustrating the obtained results to ensure their validity and applicability.