Walks through dimensions by positive definid functions

• Grant number: 21/04269-0
• Support type: Regular Research Grants
• Duration: November 01, 2021 - October 31, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal researcher: Ana Paula Peron
• Grantee: Ana Paula Peron
• Home 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: walks through the dimensions by positive definite functions and characterization of positive definite functions. The walks through the dimensions are related to certain differential and integral operators, usually called Descente and Montée, which preserve the positive definiteness of the functions changing the dimension of the space where such functions were definite positive.Beatson & zu Castell and Bingham & Symons provide results when the space is, respectively, the d-dimensional real sphere S^d and the product S^dxR. Matheron provides results when the space is the k-dimensional Euclidean space R^k. Our purpose is obtained operators of Descente and Montée which preserve the positive definiteness of the functions definite positive on the hypertorii T^{d_1,d_2}=S^{d_1}xS^{d_2} and on generalized spaces S^dxR^k. In addition, we intend to revisit Matheron's results in the spherical context. In his well-known paper, Schoenberg provides the characterization of positive definite functions on S^d. The search for the complete characterization of positive defined functions in different spaces has been a very current line of research in which several researchers have been engaged. In this context we propose to obtain the characterization of positive defined functions on T^{d_1, d_2}xR^k and to provide a constructive criterion that allows the construction of new classes of parametric functions that are defined positive on T^{d_1 , d_2}xR^k. (AU)