Isotropic positive definite functions on spheres play important roles inspatial statistics, where they occur as the correlation functions ofhomogeneous random fields and star-shaped random particles. In approximationtheory, strictly positive definite functions serve as radial basis functionsfor interpolating scattered data on spherical domains. We reviewcharacterizations of positive definite functions on spheres in terms ofGegenbauer expansions and apply them to dimension walks, where monotonicityproperties of the Gegenbauer coefficients guarantee positive definiteness inhigher dimensions. Subject to a natural support condition, isotropic positivedefinite functions on the Euclidean space $\mathbb{R}^3$, such as Askey's andWendland's functions, allow for the direct substitution of the Euclideandistance by the great circle distance on a one-, two- or three-dimensionalsphere, as opposed to the traditional approach, where the distances aretransformed into each other. Completely monotone functions are positivedefinite on spheres of any dimension and provide rich parametric classes ofsuch functions, including members of the powered exponential, Mat\'{e}rn,generalized Cauchy and Dagum families. The sine power family permits acontinuous parameterization of the roughness of the sample paths of a Gaussianprocess. A collection of research problems provides challenges for future workin mathematical analysis, probability theory and spatial statistics.
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