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This paper presents a parametric family of compactly-supported positivesemidefinite kernels aimed to model the covariance structure of second-orderstationary isotropic random fields defined in the $d$-dimensional Euclideanspace. Both the covariance and its spectral density have an analytic expressioninvolving the hypergeometric functions ${}_2F_1$ and ${}_1F_2$, respectively,and four real-valued parameters related to the correlation range, smoothnessand shape of the covariance. The presented hypergeometric kernel familycontains, as special cases, the spherical, cubic, penta, Askey, generalizedWendland and truncated power covariances and, as asymptotic cases, theMat\'ern, Laguerre, Tricomi, incomplete gamma and Gaussian covariances, amongothers. The parameter space of the univariate hypergeometric kernel isidentified and its functional properties -- continuity, smoothness, transitiveupscaling (mont\'ee) and downscaling (descente) -- are examined. Several setsof sufficient conditions are also derived to obtain valid stationary bivariateand multivariate covariance kernels, characterized by four matrix-valuedparameters. Such kernels turn out to be versatile, insofar as the direct andcross-covariances do not necessarily have the same shapes, correlation rangesor behaviors at short scale, thus associated with vector random fields whosecomponents are cross-correlated but have different spatial structures.

The Gauss Hypergeometric Covariance Kernel for Modeling Second-Order  Stationary Random Fields in Euclidean Spaces: its Compact Support, Properties  and Spectral Representation