A function \(f:\,{\mathbb S}^d\times{\mathbb S}^d\to{\mathbb R}\) is called positive definite if \[ \sum_{i=1}^n\sum_{j=1}^n c_i c_j f\left(u_i,u_j\right)\geq 0 \] for all \(u_1,\dots,u_n\in{\mathbb S}^d\), and \(c_1,\dots,c_n\in{\mathbb R}\). It is called isotropic if there exists a function \(\bar f:\,[0,\pi]\to{\mathbb R}\), such that \[ f(u,v)=\bar f\left(\theta\left(u,v\right)\right),\quad\forall u,v\in{\mathbb S}^d, \] where \(\theta\) is the geodesic distance on \({\mathbb S}^d\) given by \(\theta\left(u,v\right)=\arccos\left(\left<u,v\right>\right)\), and \(\left<\cdot,\cdot\right>\) is the usual inner product on \({\mathbb R}^{d+1}\).
Let \(\Psi_d\) denote the class of continuous functions \(\psi:\,[0,\pi]\to{\mathbb R}\) with \(\psi(0)=1\), such that the isotropic function \(\psi\left(\theta\left(\cdot,\cdot\right)\right)\) is positive definite.
The spherical convolution of two isotropic functions is given by \[ \left(f\circledast g\right)(u,v)=\int_{{\mathbb S}^d}\bar f\left(\theta\left(u,w\right)\right) \bar g\left(\theta\left(w,v\right)\right)\,dw,\quad u,v\in{\mathbb S}^d, \] where the integration is taken with respect to the \(d\)-dimensional Hausdorff measure on \({\mathbb S}^d\). “A function \(\varphi:\,[0,\pi]\to{\mathbb R}\) has a spherical convolution root if there exists an isotropic function \(g:\,{\mathbb S}^d\times{\mathbb S}^d\to{\mathbb R}\) such that \(\varphi\left(\theta\left(\cdot,\cdot\right)\right)=g\circledast g\).”
It is an interesting question which functions have spherical convolution roots. The author gives a sufficient condition:
“{ Theorem 1.1} Any \(\psi\in\Psi_d\) has a spherical convolution root which can be taken to be real-valued and isotropic.”
Another main result of the article answers the question on differentiability of functions from the classes \(\Psi_d\). It is known that a radial positive definite function in \({\mathbb R}^d\) has a continuous derivative of order \(\left[(d-1)/2\right]\) see [I. J. Schoenberg, Ann. Math. (2) 39, 811–841 (1938; Zbl 0019.41503)]. The author proves results confirming the conjecture of T. Gneiting [Bernoulli 19, No. 4, 1327–1349 (2013; Zbl 1283.62200)] about the same conclusion for spheres. Namely, the following theorem holds true.
“{ Theorem 1.2.} The functions in the class \(\Psi_d\) admit a continuous derivative of order \(\left[(d-1)/2\right]\) on the open interval \((0,\pi)\).”
As the author notes, the derivatives at \(0\) can be finite or infinite. For the other endpoint, \(\pi\), the author conjectured the same, but was unable to give an example.
Gegenbauer polynomials and Gegenbauer coefficients are widely used throughout the article. The paper also contains several interesting relations on the spherical convolutions. As the author notes, Theorem 1.1 also has important applications in statistics. An extensive bibliography outlines several important results obtained in the area.
The article should be interesting for specialists in harmonic analysis, polynomials, special functions, and statistics.