Positive definite functions

• Grant number: 16/03015-7
• Support Opportunities: Scholarships abroad - Research
• Start date: July 01, 2016
• End date: January 31, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Ana Paula Peron
• Grantee: Ana Paula Peron
• Host Investigator: Emilio Porcu
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: Universidad Técnica Federico Santa María (USM), Chile

Abstract

Recently, Guella & Menegatto & Peron and Berg & Porcu obtained results about the characterization of continuous, positive definite, complex valued kernels defined in the product of two real spheres and in the product of real spheres with a locally compact group. Results about strict positive definiteness were also proved for some of these kernels and for matrix valued functions defined in the product of two real spheres (Bonfim & Menegatto, Guella & Menegatto, Guella & Menegatto & Peron).In this project we propose to obtain new results about the characterization of positive definite functions, in several contexts:We will consider continuous complex valued kernels in the product of complex spheres with a locally compact group, and also continuous matrix valued functions defined in the product of real spheres with a locally compact group.We also propose to find a characterization of strict positive definiteness in these and some other cases.In more details, we would like to solve the following problems:1. characterizing the positive definite kernels in the product of complex spheres with a locally compact group;2. characterizing, by exploiting the above result, the strict positive definiteness of such kernels;3. characterizing the positive definite, matrix valued functions defined in the product of real spheres with a locally compact group;4. characterizing the strictly positive definite kernels in the product of real spheres with a locally compact group. With particular interest, for the cases in which the group is the circle or, because of its applications, the group is the real line.

Scientific publications

(7)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

BERG, CHRISTIAN; PERON, ANA P.; PORCU, EMILIO. Orthogonal expansions related to compact Gelfand pairs. EXPOSITIONES MATHEMATICAE, v. 36, n. 3-4, SI, p. 259-277, 2018. (16/03015-7, 14/25796-5)

MASSA, EUGENIO; PERON, ANA PAULA; PORCU, EMILIO. Positive Definite Functions on Complex Spheres and their Walks through Dimensions. Symmetry Integrability and Geometry-Methods and Applications, v. 13, 2017. (14/25398-0, 14/25796-5, 16/03015-7)

BERG, CHRISTIAN; PERON, ANA P.; PORCU, EMILIO. Orthogonal expansions related to compact Gelfand pairs. EXPOSITIONES MATHEMATICAE, v. 36, n. 3-4, p. 19-pg., 2018-01-01. (16/03015-7, 14/25796-5)

PERON, ANA; PORCU, EMILIO; EMERY, XAVIER. Admissible nested covariance models over spheres cross time. STOCHASTIC ENVIRONMENTAL RESEARCH AND RISK ASSESSMENT, v. 32, n. 11, p. 3053-3066, NOV 2018. (16/03015-7, 16/09906-0)

CASTRO, MARIO H.; MASSA, EUGENIO; PERON, ANA PAULA. Characterization of Strict Positive Definiteness on products of complex spheres. POSITIVITY, v. 23, n. 4, p. 853-874, SEP 2019. (16/03015-7, 14/25796-5, 14/25398-0, 16/09906-0)

BERG, CHRISTIAN; PERON, ANA P.; PORCU, EMILIO. Schoenberg's theorem for real and complex Hilbert spheres revisited. Journal of Approximation Theory, v. 228, p. 58-78, APR 2018. (16/03015-7, 14/25796-5, 16/09906-0)

GUELLA, JEAN C.; MENEGATTO, VALDIR A.; PERON, ARIA P.. Strictly Positive Definite Kernels on a Product of Spheres II. Symmetry Integrability and Geometry-Methods and Applications, v. 12, 2016. (16/03015-7, 14/25796-5, 14/00277-5)