Schwartz函数、Hadamard乘积和Dixmier-Malliavin定理
Schwartz functions, Hadamard products, and the Dixmier-Malliavin theorem
摘要：
本文证明了$\prod_{n\ge1}\frac{1}{\left(1+\frac{x^{2}}{a_{n}^{2}}\right)}$形式的函数，其中$a_{n}>0$和$\sum_{n\ge1}\frac{1}{a_{n}^{2}}<\infty$在实直线的施瓦兹空间中，回答了Casselman提出的一个问题.因此，我们得到了大量的简化，在证明的Dixlane和Malliavin的定理，每个测试功能的李群是一个有限的线性组合的卷积的两个检验函数，和类似的Fréchet空间李群表示。
英文摘要：
In this paper we show that functions of the form $\prod_{n\ge1}\frac{1}{\left(1+\frac{x^{2}}{a_{n}^{2}}\right)}$ where $a_{n}>0$ and $\sum_{n\ge1}\frac{1}{a_{n}^{2}}<\infty$ are in the Schwartz space of the real line, answering a question raised by Casselman. As a consequence we obtain substantial simplifications in the proofs of Dixmier and Malliavin of their theorem that every test function on a Lie group is a finite linear combination of convolutions of two test functions, and an analogue of this for Fréchet space Lie group representations.
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