(Strict) positive definite functions and differentiability

• Grant number: 14/25796-5
• Support Opportunities: Regular Research Grants
• Start date: March 01, 2015
• End date: April 30, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Ana Paula Peron
• Grantee: Ana Paula Peron
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The aim of this project is to obtain new results related to two lines of research: differentiability of positive definite functions (PD) and characterization of strictly positive definite functions (SPD). Recently, Buescu and Paixão ([BP11]) proved that given a PD function in $\mathbb{R}$, if some even order derivative in the origin is zero, then the function is constant, and they also obtained sufficient conditions on the even order derivatives in the origin for the function to be analytic, producing the maximal stripe where it can be holomorphically extended. In [BP14], analogous results were obtained for the case of PD functions in $\mathbb{C}$.Our purpose is to extend the results of Buescu and Paixão from [BP11, BP14] to the cases $\mathbb{R}^m$ and $\mathbb{C}^m$, $m\geq2$. About our second line of research, Chen, Menegatto and Sun ([CMS03]) obtained a full characterization of the SPD functions in $S^{m-1}$, $m\geq2$, in the classical case (scalar functions). Here we propose to obtain a characterization for the PD and SPD functions, in different settings. (AU)

Scientific publications

(9)

(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)

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)

GUELLA, J. C.; MENEGATTO, V. A.; PERON, A. P.. AN EXTENSION OF A THEOREM OF SCHOENBERG TO PRODUCTS OF SPHERES. Banach Journal of Mathematical Analysis, v. 10, n. 4, p. 671-685, OCT 2016. (12/22161-3, 14/25796-5, 14/00277-5)

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. 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)

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)

MASSA, E.; PERON, A. P.; PIANTELLA, A. C.. Estimates on the derivatives and analyticity of positive definite functions on R-m. ANALYSIS MATHEMATICA, v. 43, n. 1, p. 89-98, MAR 2017. (14/25796-5, 14/25398-0)

GUELLA, J. C.; MENEGATTO, V. A.; PERON, A. P.. Strictly positive definite kernels on a product of circles. POSITIVITY, v. 21, n. 1, p. 329-342, MAR 2017. (12/22161-3, 14/25796-5, 14/00277-5)

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)