Differentiability in reproducing Kernel Hilbert space on the sphere

Author(s)

Thaís Jordão

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2012-03-02

Examining board members

Valdir Antonio Menegatto; Cleonice Fatima Bracciali; Ana Paula Peron; Fernando Rodrigo Rafaeli; Eduardo Brandani da Silva

• Advisor: Valdir Antonio Menegatto

Abstract

A reproducing kernel Hilbert space (EHR) is a Hilbert space of functions constructed in a unique manner from a fixed positive definite generating kernel. The values of a function in a reproducing kernel Hilbert space can be reproduced through an elementary operation involving the function itself, the generating kernel and the inner product of the space. In this work, we consider reproducing kernel Hilbert spaces generated by a positive definite kernel on the usual m-dimensional sphere. The main goal is to analyze differentiability properties inherited by the functions in the space when the generating kernel carries a differentiability assumption. That is done in two different cases: using the usual notion of differentiability on the sphere and using another one defined through multiplicative operators. The second case includes the Laplace-Beltrami derivative and fractional derivatives as well. In both cases we consider specific properties of the embeddings of the reproducing kernel Hilbert space into spaces of smooth functions induced by notion of differentiability used (AU)