A refined variant of Hartley convolution: algebraic structures, spectral radius and related issues

In this work, we propose a novel convolution product associated with the H-transform, denoted by ∗H, and explore its fundamental properties. Here, the H-transform may be regarded as a refined variant of the classical Fourier, Hartley transform, with kernel function depending on two parameters a, b. Our first contribution shows that the space of integrable functions, equipped with multiplication given by the ∗H-convolution, constitutes the commutative Banach algebra over the complex field, albeit without an identity element. Second, establishes the Wiener–Lévy type invertibility criterion for H-algebras, obtained through the density property and process of unitarization, which serves as a key step toward the proof of Gelfand's spectral radius theorem. Third, provides an explicit upper-bound of Young's inequality for ∗H-convolution and its direct corollary. Finally, all of these theoretical findings are applied to analyse specific classes of the Fredholm integral equations and heat source problems, yielding a priori estimates under the established assumptions.