Schwartz functions, Hadamard products, and the Dixmier-Malliavin theorem

In this paper we show that functions of the form ${\prod }_{n\ge 1}\frac{1}{(1+\frac{x^2}{a_n^2})}$ where $a_n>0$ and ${\sum }_{n\ge 1}\frac{1}{a_n^2}<\mathrm{\infty }$ 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.