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

Twisted convolution and Moyal star product of generalized functions