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Doctoral Thesis
DOI
10.11606/T.55.2011.tde-14092011-094712
Document
Author
Full name
Mario Henrique de Castro
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2011
Supervisor
Committee
Menegatto, Valdir Antonio (President)
Fávaro, Vinícius Vieira
Fu, Ma To
Pellegrino, Daniel Marinho
Tozoni, Sergio Antonio
Title in Portuguese
Decaimento dos autovalores de operadores integrais positivos gerados por núcleos Laplace-Beltrami diferenciáveis
Keywords in Portuguese
Autovalores
Decaimento
Derivada de Laplace-Beltrami
Operadores integrais
Operadores positivos
Valores singulares
Abstract in Portuguese
Neste trabalho obtemos taxas de decaimento para autovalores e valores singulares de operadores integrais gerados por núcleos de quadrado integrável sobre a esfera unitária em 'R POT. m+1', m 2, sob hipóteses sobre ambos, certas derivadas do núcleo e o operador integral gerado por tais derivadas. Este tipo de problema é comum na literatura, mas as hipóteses geralmente são definidas via diferenciação usual em 'R POT m+1'. Aqui, as hipóteses são todas definidas via derivada de Laplace-Beltrami, um conceito genuinamente esférico investigado primeiramente por W. Rudin no começo dos anos 50. As taxas de decaimento apresentadas são ótimas e dependem da dimensão m e da ordem de diferenciabilidade usada para definir as condições de suavidade
Title in English
Eigenvalue decay of positive integral operators generated by Laplace-Beltrami differentiable kernels
Keywords in English
Decay rates
Eigenvalues
Integral operators
Laplace-Beltrami derivative
Positive operators
Singular numbers
Abstract in English
In this work we obtain decay rates for singular values and eigenvalues of integral operators generated by square integrable kernels on the unit sphere in 'R m+1', m 2, under assumptions on both, certain derivatives of the kernel and the integral operators generated by such derivatives. This type of problem is common in the literature but the assumptions are usually defined via standard differentiation in 'R POT. m+1'. Here, the assumptions are all defined via the Laplace-Beltrami derivative, a concept first investigated by W. Rudin in the early fifties and genuinely spherical in nature. The rates we present are optimal and depend on both, the differentiability order used to define the smoothness conditions and the dimension m
 
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Publishing Date
2011-09-14
 
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