Cosine Manifestations of the Gelfand Transform

The goal of the paper is to provide a detailed explanation on how the (continuous) cosine transform and the discrete(-time) co-sine transform arise naturally as certain manifestations of the celebrated Gelfand transform. We begin with the introduction of the cosine convolution *(c), which can be viewed as an arithmetic mean of the classical convolution and its twin brother, the anticonvolution. D'Alembert's property of *(c) plays a pivotal role in establishing the bijection between Delta(L-1(G), *(c)) and the cosine class COS( G), which turns out to be an open map if COS( G) is equipped with the topology of uniform convergence on compacta t(ucc). Subsequently, if G = R, Z, S-1 or Z(n) we find a relatively simple topological space which is homeomorphic to Delta(L-1 (G), *(c)). Finally, we witness the reduction of the Gelfand transform to the aforementioned cosine transforms.

Automated summary

This paper provides a detailed explanation on how the (continuous) cosine transform and the discrete(-time) co-sine transform naturally arise as manifestations of the celebrated Gelfand transform. The introduction of the cosine convolution *(c) plays a key role in establishing the bijection between Delta(L-1(G), *(c)) and the cosine class COS( G), and in reducing the Gelfand transform to the cosine transforms.