Inequalities for differentiable reproducing kernels and an application to positive integral operators
© J. Buescu and A. C. Paix˜ao. 2006
Received: 18 October 2005
Accepted: 13 November 2005
Published: 3 May 2006
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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