We present a characterization for the continuous, isotropic, and positive definite kernels on a product of spheres along the lines of a classical result of Schoenberg on positive definiteness on a single sphere. We also discuss a few issues regarding the characterization, including topics for future investigation.
J. C. Guella. V. A. Menegatto. A. P. Peron. "An extension of a theorem of Schoenberg to products of spheres." Banach J. Math. Anal. 10 (4) 671 - 685, October 2016. https://doi.org/10.1215/17358787-3649260