Cosine manifestations of the Gelfand transform

TL;DR

This paper explores how the continuous and discrete cosine transforms naturally emerge as manifestations of the Gelfand transform, using the cosine convolution and topological analysis to establish their relationships.

Contribution

It introduces the cosine convolution and demonstrates its role in connecting the Gelfand transform with cosine transforms for various groups, providing a unified theoretical framework.

Findings

The cosine convolution $igstar_c$ acts as an arithmetic mean of classical convolution and anticonvolution.

A bijection between the Gelfand spectrum and the cosine class is established and shown to be an open map.

The Gelfand transform reduces to cosine transforms for specific groups like $ eal$, $z$, $S^1$, and $z_n$.

Abstract

The goal of the paper is to provide a detailed explanation on how the (continuous) cosine transform and the discrete(-time) cosine transform arise naturally as certain manifestations of the celebrated Gelfand transform. We begin with the introduction of the cosine convolution $\star_c$, which can be viewed as an "arithmetic mean" of the classical convolution and its "twin brother", the anticonvolution. The d'Alambert property of $\star_c$ plays a pivotal role in establishing the bijection between $\Delta(L^1(G),\star_c)$ and the cosine class $\mathcal{COS}(G),$ which turns out to be an open map if $\mathcal{COS}(G)$ is equipped with the topology of uniform convergence on compacta $\tau_{ucc}$. Subsequently, if $G = \mathbb{R},\mathbb{Z}, S^1$ or $\mathbb{Z}_n$ we find a relatively simple topological space which is homeomorphic to $\Delta(L^1(G),\star_c).$ Finally, we witness the "reduction" of the Gelfand transform to the aforementioned cosine transforms.