Twisted convolution and Moyal star product of generalized functions

TL;DR

This paper explores the properties of twisted convolution and Moyal star product within nuclear function spaces, extending the Weyl symbol calculus beyond tempered distributions.

Contribution

It characterizes the algebra of convolution multipliers and describes the Moyal multiplier algebra in a generalized functional analysis setting.

Findings

The algebra of convolution multipliers includes all rapidly decreasing dual space functionals.

The Moyal multiplier algebra is explicitly characterized for Fourier-transformed spaces.

The results extend Weyl symbol calculus to broader classes of distributions.

Abstract

We consider nuclear function spaces on which the Weyl-Heisenberg group acts continuously and study the basic properties of the twisted convolution product of the functions with the dual space elements. The final theorem characterizes the corresponding algebra of convolution multipliers and shows that it contains all sufficiently rapidly decreasing functionals in the dual space. Consequently, we obtain a general description of the Moyal multiplier algebra of the Fourier-transformed space. The results extend the Weyl symbol calculus beyond the traditional framework of tempered distributions.