Positive definite matrices and differentiable reproducing kernel inequalities

Let I ⊆ R be a interval and k : I 2 →C 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. © 2005 Elsevier Inc. All rights reserved