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  • Research Article
  • Open Access

Inequalities for differentiable reproducing kernels and an application to positive integral operators

Journal of Inequalities and Applications20062006:53743

https://doi.org/10.1155/JIA/2006/53743

  • Received: 18 October 2005
  • Accepted: 13 November 2005
  • Published:

Abstract

Let be an interval and let be a reproducing kernel on . We show that if is in the appropriate differentiability class, it satisfies a 2-parameter family of inequalities of which the diagonal dominance inequality for reproducing kernels is the 0th order case. We provide an application to integral operators: if is a positive definite kernel on (possibly unbounded) with differentiability class and satisfies an extra integrability condition, we show that eigenfunctions are and provide a bound for its Sobolev norm. This bound is shown to be optimal.

Keywords

  • Integral Operator
  • Integrability Condition
  • Sobolev Norm
  • Order Case
  • Diagonal Dominance

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Authors’ Affiliations

(1)
Departamento de Matemática, Instituto Superior Técnico, Lisbon, 1049-001, Portugal
(2)
Departamento de Engenharia Mecânica, ISEL, Lisbon, 1949-014, Portugal

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