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Complex Positive Definite Functions on Strips

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Abstract

We characterize a holomorphic positive definite function f defined on a horizontal strip of the complex plane as the Fourier–Laplace transform of a unique exponentially finite measure on \({\mathbb R}\). With this characterization, the classical theorems of Bochner on positive definite functions and of Widder on exponentially convex functions become respectively the real and imaginary sections of the corresponding complex integral representation. We provide minimal holomorphy assumptions for this characterization and derive conclusions for meromorphic functions under minimal positive definiteness conditions. Further characterizations are derived from conditions on the derivatives of f arising from the study of the usual concepts of moment, moment-generating function and characteristic function in this context.

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Author information

Authors and Affiliations

  1. Dep. Matemática, FCUL and CMAF, Lisbon, Portugal

    Jorge Buescu

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

    A. C. Paixão

  3. Dep. Matemática, FCUL, Lisbon, Portugal

    A. Symeonides

Authors
  1. Jorge Buescu
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  2. A. C. Paixão
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  3. A. Symeonides
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Corresponding author

Correspondence to Jorge Buescu.

Additional information

Communicated by Igor Klep.

The first author acknowledges partial support by Fundação para a Ciência e Tecnologia, UID/MAT/04561/2013.

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Buescu, J., Paixão, A.C. & Symeonides, A. Complex Positive Definite Functions on Strips. Complex Anal. Oper. Theory 11, 627–649 (2017). https://doi.org/10.1007/s11785-015-0527-y

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