From Schwartz space to Mellin transform

TL;DR

This paper explores the natural emergence of the Mellin transform from the structure of Schwartz functions on positive reals, establishing its algebraic properties and analogy to the Gelfand transform.

Contribution

It provides a detailed analysis of Schwartz functions on positive reals, introduces the Mellin convolution algebra, and demonstrates the Mellin transform's connection to the Gelfand transform.

Findings

The space of Schwartz functions forms a commutative Fréchet algebra under Mellin convolution.

The structure space of this algebra is homeomorphic to the real line.

The Mellin transform can be derived similarly to the Gelfand transform construction.

Abstract

The primary motivation behind this paper is an attempt to provide a thorough explanation of how the Mellin transform arises naturally in a process akin to the construction of the celebrated Gelfand transform. We commence with a study of a class of Schwartz functions $\mathcal{S}(\mathbb{R}_+),$ where $\mathbb{R}_+$ is the set of all positive real numbers. Various properties of this Fr\'echet space are established and what follows is an introduction of the Mellin convolution operator, which turns $\mathcal{S}(\mathbb{R}_+)$ into a commutative Fr\'echet algebra. We provide a simple proof of Mellin-Young convolution inequality and go on to prove that the structure space $\Delta(\mathcal{S}(\mathbb{R}_+),\star)$ (the space of nonzero, linear, continuous and multiplicative functionals $m:\mathcal{S}(\mathbb{R}_+)\longrightarrow \mathbb{R}$) is homeomorphic to $\mathbb{R}.$ Finally, we show that the Mellin transform arises in a process which bears a striking resemblance to the construction of the Gelfand transform.