In search of a non-Archimedean Schwartz space

We point out the problems of finding a good substitute for Schwartz space in a non-Archimedean setting and present a "No-go" theorem. Given these problems, we explore the possibilities of using the multiplicative structure of a local field (instead of the additive structure) and present some results in that direction. Our space of test functions is constructed from a family of weighted modulation spaces (Feichtinger algebras).