Reproducing Properties of Differentiable Mercer-Like Kernels on the Sphere

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We study differentiability of functions in the reproducing kernel Hilbert space (RKHS) associated with a smooth Mercer-like kernel on the sphere. We show that differentiability up to a certain order of the kernel yields both, differentiability up to the same order of the elements in the series representation of the kernel and a series representation for the corresponding derivatives of the kernel. These facts are used to embed the RKHS into spaces of differentiable functions and to deduce reproducing properties for the derivatives of functions in the RKHS. We discuss compactness and boundedness of the embedding and some applications to Gaussian-like kernels.