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

TL;DR

This paper demonstrates that certain infinite product functions are Schwartz functions, simplifying the proof of the Dixmier-Malliavin theorem on test functions on Lie groups and extending it to Fréchet space representations.

Contribution

It proves that specific infinite product functions belong to the Schwartz space, leading to simplified proofs of the Dixmier-Malliavin theorem and its extensions.

Findings

Infinite product functions are in the Schwartz space.

Simplified proof of the Dixmier-Malliavin theorem.

Extended the theorem to Fréchet space Lie group representations.

Abstract

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.