We recognize a result of Schreiner, concerning strictly positive definite functions on a sphere in an Euclidean space, as a generalization of Bochner's theorem for compact groups.
Allali, Mohamed, and Tomasz Przebinda. "Strictly positive definite functions on a compact group." Proceedings of the American Mathematical Society (2001): 1459-1462. doi: 10.1090/S0002-9939-00-05687-2
American Mathematical Society