论文标题
从施瓦茨空间到梅林变换
From Schwartz space to Mellin transform
论文作者
论文摘要
本文背后的主要动机是试图对梅林变换的方式进行详尽的解释，这在类似于著名的Gelfand Transform的过程中自然而然地产生。我们从研究一类Schwartz函数的研究开始，$ \ Mathcal {s}（\ Mathbb {r} _+），$，其中$ \ mathbb {r} _+$是所有正实际数字的集合。建立了此Fréchet空间的各种属性，接下来是Mellin卷积运算符的介绍，该操作员将$ \ Mathcal {s}（\ Mathbb {r} _+）$变成了交换性fréchet代数。我们提供了一个简单的Mellin-Young卷积不平等的证明，并继续证明结构空间$δ（\ Mathcal {s}（\ Mathbb {r} _+），\ star），\ star）$（非零，线性，连续，连续和乘函数的空间\ mathbb {r} $）与$ \ mathbb {r}。$最终，我们证明了梅林变换在一个过程中出现，与gelfand变换的构建具有惊人的相似之处。
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 Fréchet space are established and what follows is an introduction of the Mellin convolution operator, which turns $\mathcal{S}(\mathbb{R}_+)$ into a commutative Fréchet algebra. We provide a simple proof of Mellin-Young convolution inequality and go on to prove that the structure space $Δ(\mathcal{S}(\mathbb{R}_+),\star)$ (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.
