Parseval–Goldstein Identities and Abelian Theorems for the Hartley Transform over Distributions of Compact Support and Generalized Functions

This work develops an analytical framework for the Hartley transform, which, unlike the Fourier transform, converts real-valued functions into real-valued functions. We derive Parseval–Goldstein-type identities for suitable classes of functions and establish Abelian theorems in the setting of compactly supported distributions and generalized functions. Moreover, a finite formulation of the Hartley transform is constructed and analyzed within the spaces of distributions of compact support and generalized functions.