Characterization of ultradifferentiable test functions defined by weight matrices in terms of their Fourier Transform

We prove that functions with compact support in non-quasianalytic classes E{M}of Roumieu-type and E(M)of Beurling-type defined by a weight matrix M with some mild regularity conditions can be characterized by the decay properties of their Fourier transform. For this we introduce the abstract technique of constructing from M multi-index matrices and associated function spaces. We study the behaviour of this construction in detail and characterize its stability. Moreover non-quasianalyticity of the classes E{M}and E(M)is characterized.