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 Fr\'echet space Lie group representations.
Categories: math.RT math.NT