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Positive-definiteness and integral representations for special functions

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Abstract

It is known that a holomorphic positive definite function f defined on a horizontal strip of the complex plane may be characterized as the Fourier–Laplace transform of a unique exponentially finite measure on \({{\mathbb {R}}}\). In this paper we apply this complex integral representation to specific families of special functions, including the \(\Gamma \), \(\zeta \) and Bessel functions, and construct explicitly the corresponding measures, thus providing new insight into the nature of complex positive and co-positive definite functions. In the case of the \(\zeta \) function this process leads to a new proof of an integral representation on the critical strip.

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Authors and Affiliations

  1. Department of Matemática, FCUL and CMAFCIO, Lisbon, Portugal

    J. Buescu

  2. Área Departamental de Matemática, ISEL, Lisbon, Portugal

    A. C. Paixão

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  1. J. Buescu
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  2. A. C. Paixão
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Correspondence to J. Buescu.

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The first author acknowledges partial support by Fundação para a Ciência e Tecnologia, UID/MAT/04561/2019.

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Buescu, J., Paixão, A.C. Positive-definiteness and integral representations for special functions. Positivity 25, 731–750 (2021). https://doi.org/10.1007/s11117-020-00784-4

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