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On positive functions with positive fourier transforms

Abstract : Using the basis of Hermite–Fourier functions (i.e. the quantum oscillator eigenstates) and the Sturm theorem, we derive constraints for a function and its Fourier transform to be both real and positive. We propose a constructive method based on the algebra of Hermite polynomials. Applications are extended to the 2-dimensional case (i.e. Fourier–Bessel transforms and the algebra of Laguerre polynomials) and to adding constraints on derivatives, such as monotonicity or convexity.
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B.G. Giraud, R. Peschanski. On positive functions with positive fourier transforms. Acta Physica Polonica B, Jagellonian University, Cracow, 2006, 37 (2), pp.331-346. ⟨cea-02889284⟩

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