Distributions of ℏ-positive type and applications

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Abstract
States in classical mechanics are probability measures, and their Fourier transforms are continuous functions of positive type. States in the phase‐space formulation of quantum mechanics are Wigner distribution functions, and their (symplectic) Fourier transforms have been characterized by Kastler [Commun. Math. Phys. 1, 14 (1965)] and Loupias and Miracle‐Sole [Commun. Math. Phys. 2, 31 (1966); Ann. Insto. H. Poincaré 6, 39 (1967)] as being continuous functions of ‘‘ℏ‐positive type.’’ In this paper, (Schwartz) distributions of ℏ‐positive type, are defined and studied. It is shown that if such distributions are bounded on a certain sequence of test functions, then they are symplectic Fourier transforms of Wigner distribution functions. These results, are applied to a variety of problems ranging from ones involving the quantum Liouville equation to a problem in signal analysis.

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