Positive definite matrices and differentiable reproducing kernel inequalities |
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Authors: | Jorge Buescu AC Paixão |
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Institution: | a Departamento Matemática, Inst. Sup. Técnico, Portugal b Departamento Mecânica, ISEL, Portugal |
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Abstract: | Let I⊆R be a interval and be a reproducing kernel on I. By the Moore-Aronszajn theorem, every finite matrix k(xi,xj) is positive semidefinite. We show that, as a direct algebraic consequence, if k(x,y) is appropriately differentiable it satisfies a 2-parameter family of differential inequalities of which the classical diagonal dominance is the order 0 case. An application of these inequalities to kernels of positive integral operators yields optimal Sobolev norm bounds. |
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Keywords: | Positive definite matrices Reproducing kernels Inequalities |
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