Positive Definite Functions on Complex Spheres and their Walks through Dimensions

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2017
• Main Authors: Massa, E., Peron, A.P., Porcu, E.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2017
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/149263
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Positive Definite Functions on Complex Spheres and their Walks through Dimensions / E. Massa, A.P. Peron, E. Porcu // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 49 назв. — англ.

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.