Differential and Integral Equations

Positive definite matrices and integral equations on unbounded domains

Jorge Buescu and A. C. Paixão

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The kernel of a continuous positive integral operator on an interval $I$ is a Moore matrix on $I$. We show that, under minimal differentiability assumptions, this implies that the kernel satisfies a 2-parameter family of differential inequalities. These inequalities ensure that, for unbounded $I$, the corresponding integral operator is exceptionally well behaved: it is compact and thus the eigenfunctions for its discrete spectrum have the differentiability of the kernel and satisfy sharp Sobolev bounds, the symmetric mixed partial derivatives are again kernels of positive operators and the differentiated eigenfunction series converge uniformly and absolutely. Converse results are derived.

Article information

Differential Integral Equations, Volume 19, Number 2 (2006), 189-210.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]
Secondary: 45P05: Integral operators [See also 47B38, 47G10] 47G10: Integral operators [See also 45P05]


Buescu, Jorge; Paixão, A. C. Positive definite matrices and integral equations on unbounded domains. Differential Integral Equations 19 (2006), no. 2, 189--210. https://projecteuclid.org/euclid.die/1356050524

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