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

Xavier Emery, Emilio Porcu

    Research output: Contribution to journalArticlepeer-review

    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.

    Original languageBritish English
    Article number28
    JournalComputational and Applied Mathematics
    Volume43
    Issue number1
    DOIs
    StatePublished - Feb 2024

    Keywords

    • Exponential convexity
    • Integral representations
    • Local stationarity
    • Positive definite functions
    • Rudin’s extensions

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