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

Journal of Inequalities and Applications volume 2006, Article number: 53743 (2006) Cite this article

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.

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Authors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, Lisbon, 1049-001, Portugal

    Jorge Buescu

  2. Departamento de Engenharia Mecânica, ISEL, Lisbon, 1949-014, Portugal

    A. C. Paixão

Authors
  1. Jorge Buescu
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  2. A. C. Paixão
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Corresponding author

Correspondence to Jorge Buescu.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Buescu, J., Paixão, A.C. Inequalities for differentiable reproducing kernels and an application to positive integral operators. J Inequal Appl 2006, 53743 (2006). https://doi.org/10.1155/JIA/2006/53743

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

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