Spaces of type $S$ and deformation quantization

We study the properties of the Gelfand–Shilov spaces $S^{b_n}_{a_k}$ in the context of deformation quantization. Our main result is a characterization of their corresponding multiplier algebras with respect to a twisted convolution, which is given in terms of the inclusion relation between these algebras and the duals of the spaces of pointwise multipliers with an explicit description of these functional spaces. The proof of the inclusion theorem essentially uses the equality $S^{b_n}_{a_k}=S^{b_n}\cap S_{a_k}$.