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Integral representations, extension theorems and walks through dimensions under radial exponential convexity

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Abstract

We consider the class of radial exponentially convex functions defined over n-dimensional balls with finite or infinite radii. We provide characterization theorems for these classes, as well as Rudin’s type extension theorems for radial exponentially convex functions defined over n-dimensional balls into radial exponentially convex functions defined over the whole n-dimensional Euclidean space. We furthermore establish inversion theorems for the measures, termed here n-Nussbaum measures, associated with integral representations of radial exponentially convex functions. This in turn allows obtaining recurrence relations between 1-Nussbaum measures and n-Nussbaum measures for a given integer n greater than 1. We also provide a up to now unknown catalogue of radial exponentially convex functions and associated n-Nussbaum measures. We finally turn our attention into componentwise radial exponential convexity over product spaces, with a Rudin extension result and analytical examples of exponentially convex functions and associated Nussbaum measures. As a byproduct, we obtain a parametric model of nonseparable stationary space-time covariance functions that do not belong to the well-known Gneiting class.

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Acknowledgements

This paper is based upon work supported by the National Agency for Research and Development of Chile under grants ANID FONDECYT 1210050 and ANID PIA AFB230001 (X. Emery), and by the Khalifa University of Science and Technology under Award No. FSU-2021-016 (E. Porcu). The authors are grateful to the anonymous reviewers, the Associate Editor for their careful reading of the paper.

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Authors and Affiliations

  1. Department of Mining Engineering & Advanced Mining Technology Center, Universidad de Chile, Santiago, Chile

    Xavier Emery

  2. Department of Mathematics, Khalifa University and ADIA Lab, Abu Dhabi, United Arab Emirates

    Emilio Porcu

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  1. Xavier Emery
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  2. Emilio Porcu
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Correspondence to Emilio Porcu.

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Communicated by Jose Alberto Cuminat.

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Emery, X., Porcu, E. Integral representations, extension theorems and walks through dimensions under radial exponential convexity. Comp. Appl. Math. 43, 28 (2024). https://doi.org/10.1007/s40314-023-02529-x

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