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      Schoenberg coefficients on real and complex spheres: towards a complete picture

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          Abstract

          Positive definite functions on spheres have received an increasing interest in many branches of mathematics and statistics. In particular, the Schoenberg sequences in the spectral representation of positive definite functions have been challenged by several mathematicians in the last years. This paper completes the picture about positive definite functions on real as well as complex spheres by providing a complete set of relations between Schoenberg sequences defined over real as well as complex spheres of different dimensions. We illustrate our findings describing an application to strict positive definiteness.

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          Positive definite functions on spheres

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            Strictly and non-strictly positive definite functions on spheres

            Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space \(\mathbb{R}^3\), such as Askey's and Wendland's functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Mat\'{e}rn, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.
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              Spherical and Hyperbolic Fractional Brownian Motion

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                Author and article information

                Journal
                21 July 2018
                Article
                1807.08184
                2e7bc3ff-7300-4d15-baec-934472ab76fc

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                17 pages
                math.CA

                Mathematics
                Mathematics

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