Positive definite and isotropic kernels on compact two-point homogeneous spaces

Author(s)

Rafaela Neves Bonfim

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2017-07-25

Examining board members

Valdir Antonio Menegatto; Vinícius Vieira Fávaro; Claudemir Pinheiro de Oliveira; Alagacone Sri Ranga

• Advisor: Valdir Antonio Menegatto

Abstract

In this work we present a characterization for the continuous, isotropic and positive definite matrix-valued kernels on a compact two-point homogeneous space. After that, we consider the strict positive definiteness of the kernels, describing some independent sufficient conditions for that property to hold. In the case the space is not a sphere, one of the conditions becomes necessary and sufficient for the strict positive definiteness of the kernel. Further, for 22- matrix-valued kernels on a compact two-point homogeneous space which is not a sphere, we present a characterization for the strict positive definiteness of the kernels based upon the main diagonal elements in its matrix representation. In the last part of this work, we restrict ourselves to scalar kernels and determine necessary and sufficient conditions in order that the product of two continuous, isotropic and positive definite kernels on a compact two-point homogeneous space be strictly positive definite. We also discuss the extension of this result for kernels defined on a product of a locally compact group and a high dimensional sphere. (AU)

• FAPESP's process: 14/14380-2 - Positive definite functions on spheres: the matricial and parametrical versions
• Grantee: Rafaela Neves Bonfim
• Support Opportunities: Scholarships in Brazil - Doctorate