Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Author(s)

Massa, Eugenio ^[1] ; Peron, Ana Paula ^[1] ; Porcu, Emilio ^{[2, 3]}

Affiliation

^[1] ICMC USP Sao Carlos, Dept Matemat, Caixa Postal 668, BR-13560970 Sao Carlos, SP - Brazil

^[2] Univ Newcastle, Chair Spatial Analyt Methods, Sch Math & Stat, Callaghan, NSW - Australia

^[3] Univ Tecn Federico Santa Maria, Dept Math, Ave Espana 1680, Valparaiso 230123 - Chile

• Document type: Journal article
• Source: Symmetry Integrability and Geometry-Methods and Applications; v. 13, 2017.
• Web of Science Citations: 3

Abstract

We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Montee and Descente operators as proposed by Beatson and zu Castell {[}J. Approx. Theory 2 2 1 (2017), 22-37] on the basis of the original Matheron operator {[}Les variables regionalisees et leur estimation, Masson, Paris, 1965], allow for similar walks through dimensions. We show that the Montee 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. (AU)

• FAPESP's process: 14/25398-0 - Elliptic equations and systems with several kinds of interaction with the spectrum
• Grantee: Eugenio Tommaso Massa
• Support Opportunities: Regular Research Grants
• FAPESP's process: 14/25796-5 - (Strict) positive definite functions and differentiability
• Grantee: Ana Paula Peron
• Support Opportunities: Regular Research Grants
• FAPESP's process: 16/03015-7 - Positive definite functions
• Grantee: Ana Paula Peron
• Support Opportunities: Scholarships abroad - Research