Ultrarapidly decreasing ultradifferentiable functions, Wigner distributions and density matrices

Spaces , , of ultradecreasing ultradifferentiable (or forshort, ultra-) functions, depending on a weight e^(x), are introducedin the context of quantum statistics. The corresponding coefficientspaces in the Fock basis are identified, and it is shown thatthe Hermite expansion is a tame isomorphism between these spaces.These results are used to link decay properties of density matricesto corresponding properties of the Wigner distribution.