Isotropic Positive Definite Functions on Spheres

In this paper, we investigate the relationship between positive definite functions on the unit sphere $\text{sph}$ and on the Euclidean space ${\text{RR}}^d$. For the dimension $d$ to be odd, a new technique is developed to establish the inheritance of positive (semi-)definite property from ${\text{RR}}^d$ to $\text{sph}$ and the converse. For $d=2$, it is proved that a function defined by $$f_{\text{t},\delta }(t)=(\text{t}-t{)}_{+}^{\delta },\delta \ge \text{f}d+12$$ is positive definite on the unit sphere ${\mathbb{S}}^2$ by restricting $\text{t}$ in an absolute range. Our results can verify a conjecture proposed by R.K. Beatson, W. zu Castell, Y. Xu and a sharp Pólya type criterion for positive definite functions on spheres.