Cosine manifestations of the Gelfand transform

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.