Abstract
We prove that any isotropic positive definite function on the sphere can be written as the spherical self-convolution of an isotropic real-valued function. It is known that isotropic positive definite functions on d d -dimensional Euclidean space admit a continuous derivative of order [ ( d − 1 ) / 2 ] [(d-1)/2] . We show that the same holds true for isotropic positive definite functions on spheres and prove that this result is optimal for all odd dimensions.
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