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

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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Correspondence to Jorge Buescu.

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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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Keywords

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