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Positive definite functions

Grant number: 16/03015-7
Support type:Scholarships abroad - Research
Effective date (Start): July 01, 2016
Effective date (End): January 31, 2017
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Ana Paula Peron
Grantee:Ana Paula Peron
Host: Emilio Porcu
Home Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Local de pesquisa : 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 (5)
(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)
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. Web of Science Citations: 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. Web of Science Citations: 4.
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. Web of Science Citations: 0.
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. Web of Science Citations: 3.
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. Web of Science Citations: 5.

Please report errors in scientific publications list by writing to: cdi@fapesp.br.