Functions with fixed order strict positive definiteness in the solution of interpolation problems

• Grant number: 24/01636-0
• Support Opportunities: Scholarships in Brazil - Master
• Start date: June 01, 2024
• End date: April 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Ana Paula Peron
• Grantee: Victor Cruz Borges
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

This project aims to introduce the student to the study of Spherical Analysis as well as to the investigation in one of the lines of research in this area, which strongly involves positive definite functions in real spheres. These functions appear in many applications, such as geostatistics, probability, Fourier analysis, interpolation problems.The characterization of positive definite functions on real spheres, obtained by I.J. Schoenberg in {Positive definite functions on spheres, Duke Math. J. 9 (1942)}, will be used for the main objective of the project: studying the article {V. Menegatto, Strict positive definiteness on spheres, Analysis (Munich), no. 3, 217--233 (1999)} which provides necessary conditions for positive definite functions on real spheres to be strictly positive definite of order $n\in\mathbb{Z}_+$ and necessary and sufficient conditions when $n=2 $. Such functions provide unique solutions in certain scattered data interpolation problems on spheres.It is worth mentioning that the complete characterization for strict definite positivity of order $n\neq2$ is still an open problem and, with time, the student might be able to propose a new contribution to the theme.