Hadamard products and Schwartz functions

Abstract

We show that functions of the form ∏ n ≥ 1 ( 1 + x 2 / a n 2 ) − 1 \prod _{n\ge 1}\left (1+x^{2}/a_{n}^{2}\right )^{-1} ( a n > 0 ) (a_{n}>0) are in the Schwartz space of the real line whenever the infinite product converges uniformly on compact subsets of C \mathbb {C} , answering a question of Bill Casselman. We also include a different proof due to Jean-Loup Waldspurger.