Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Abstract: We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Mont\'ee and Descente operators as proposed by Beatson and zu Castell [J. Approx. Theory 221 (2017), 22-37] on the basis of the original Matheron operator [Les variables r\'egionalis\'ees et leur estimation, Masson, Paris, 1965], allow for similar walks through dimensions. We show that the Mont\'ee operators also preserve, up to a constant, strict positive definiteness. For the Descente operators, we show that strict positive definiteness is preserved under some additional conditions, but we provide counterexamples showing that this is not true in general. We also provide a list of parametric families of (strictly) positive definite functions over complex spheres, which are important for several applications.