Many applications in spatial statistics involve data observed over large regions on the Earth's surface. There is a large literature devoted to the characterization and properties of covariance functions on Euclidean spaces, and there have been some recent advances on the characterization of covariance functions on spheres. We demonstrate that valid isotropic covariance functions on spheres may be easily specified via a Fourier series representation, and we give several examples, including an analog to the Mat{\'{e}}rn covariance function, which has been a popular choice due to its ability to specify the degree of smoothness of the process. Correctly specifying the smoothness or mean square differentiability of a process is important for spatial prediction, so we adapt some of the results on mean square differentiable processes on Euclidean spaces to spheres. In doing so, we define the notion of a mean square differentiable process on a sphere and give necessary and sufficient conditions for an isotropic covariance function on a sphere to correspond to an $m$ times mean square differentiable process. These conditions imply that if a process on a Euclidean space is restricted to a sphere of lower dimension, the process will retain its mean square differentiability properties. The restriction requires the covariance function to take Euclidean distance as its argument. To address the issue of whether covariance functions using Euclidean distance result in poorly fitting models, we compare our new analog to the Mat{\'{e}}rn, which takes great circle distance as its argument, to the usual Mat{\'{e}}rn that is only generally valid with Euclidean distance. These covariance functions are compared with several others in applications involving satellite and climate model data.
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