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.

DOI Code: 10.1285/i15900932v36n2p1

Keywords: Ultradifferentiable functions; non-quasianalyticity; Fourier transform

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