### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 088, 16 pages      arXiv:1704.01237      https://doi.org/10.3842/SIGMA.2017.088

### Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Eugenio Massa a, Ana Paula Peron a and Emilio Porcu bc
a) Departamento de Matemática, ICMC-USP - São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
b) School of Mathematics and Statistics, Chair of Spatial Analytics Methods, University of Newcastle, UK
c) Department of Mathematics, Universidad Técnica Federico Santa Maria, Avenida España 1680, Valparaíso, 230123, Chile

Received April 06, 2017, in final form October 30, 2017; Published online November 08, 2017

Abstract
We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Montée 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 régionalisées et leur estimation, Masson, Paris, 1965], allow for similar walks through dimensions. We show that the Montée 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.

Key words: Descente; disk polynomials; Montée; positive definite functions.

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