Strict positive definiteness on spheres via disk polynomials
The paper under review is concerned with zonal spherical harmonic analysis. It studies kernels on the unit sphere \(\Omega_q\) of \({\mathbb C}^q\) admitting the form \(K_f: (z,w) \to f(\langle z| w\rangle)\) where the complex-valued function \(f\) is continuous on the closed unit disk of the complex plane \({\mathbb C}\), and \(\langle.| .\rangle\) denotes the standard scalar product of the complex vector space \({\mathbb C}^q, q \geq 3\). The function \(f\) is called the shape function of \(K_f\). Specifically, the problem of characterizing kernels \(K_f\) of strict positive type is considered. The characterization obtained depends on the expansion of the shape function \(f\) in terms of suitable disk polynomials. Disk polynomials can be expressed in terms of classical Jacobi polynomials [\textit{J. N. Boyd} and \textit{P. N. Raychowdhury}, ``Zonal harmonic functions from two-dimensional analogs of Jacobi polynomials'', Appl. Anal. 16, 243--259 (1983; Zbl 0503.33009)]. The main result of the paper demonstrates that the property of \(K_f\) of being a kernel of strict positive type can be read off from the index parity \(m - n\) of strictly positive coefficients \(a_{m,n}(f), m\geq 0, n\geq 0\), derived from the expansion of \(f\) in terms of disk polynomials [\textit{V. A. Menegatto} and \textit{A. P. Peron}, ``A complex approach to strict positive definiteness on spheres'', Integral Transform., Spec. Funct. 11, 377--396 (2001; Zbl 1025.42006); ``Positive definite kernels on complex spheres'', J. Math. Anal. Appl. 254, 219--232 (2001; Zbl 0973.32004)]. An extension of the main result to the unit sphere \(\Omega_\infty\) of \(\ell^2\) is also described.
scientific article; zbMATH DE number 1821314