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 $\mathcal{E}_{\{\mathcal{M}\}}$ of Roumieu-type and $\mathcal{E}_{(\mathcal{M})}$ of Beurling-type defined by a weight matrix $\mathcal{M}$ 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 $\mathcal{M}$ 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 mathcal{E}_{\{\mathcal{M}\}}$ and $\mathcal{E}_{(\mathcal{M})}$ is characterized