Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Symmetry Integrability and Geometry-Methods and Applications

article

Eugenio Massa¹  Ana Paula Peron¹  Emilio Porcu² 

[1] Departamento de Matemática, ICMC-USP - São Carlos;Department of Mathematics, Universidad T´ecnica Federico Santa Maria

关键词: Descente;    disk polynomials;    Mont´ee;    positive definite functions;   

DOI  :  10.3842/SIGMA.2017.088

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown