Gelfand变换的余弦表示
Cosine manifestations of the Gelfand transform
摘要：
本文的目的是提供一个详细的解释如何（连续）余弦变换和离散（时间）余弦变换自然产生的某些表现形式的著名Gelfand变换。我们从介绍余弦卷积$\star_c$开始，它可以被看作是经典卷积和它的“孪生兄弟”反卷积的“算术平均“。$\star_c$的d'Alambert性质在建立$Δ(L^1(G),\star_c)$和余弦类$\mathcal{COS}(G),$之间的双射中起着关键作用，如果$\mathcal{COS}(G)$在余弦类$τ_{ucc}$上具有一致收敛拓扑，则该双射是开映射.随后，如果$G = \mathbb{R},\mathbb{Z}, S^1$或$\mathbb{Z}_n$，我们找到一个相对简单的拓扑空间，它是同胚的$Δ(L^1(G),\star_c).$最后，我们见证了“减少”的Gelfand变换上述余弦变换。
英文摘要：
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 $Δ(L^1(G),\star_c)$ 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 $τ_{ucc}$. Subsequently, if $G = \mathbb{R},\mathbb{Z}, S^1$ or $\mathbb{Z}_n$ we find a relatively simple topological space which is homeomorphic to $Δ(L^1(G),\star_c).$ Finally, we witness the "reduction" of the Gelfand transform to the aforementioned cosine transforms.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
