04/02/2005, 15:00 — 16:00 — Sala P10, Pavilhão de Matemática
Jorge Buescu, IST/DM
General Inequalities for differentiable reproducing kernels
Let $E \subseteq \mathbb{R}$ be an abstract space and $k: I^2 \to \mathbb{C}$ be a reproducing kernel on $E$. By the Moore-Aronszajn theorem, every finite matrix $k(x_i,x_j)$ is positive semidefinite. If $E= \mathbb{R}^n$ or $E= \mathbb{C}^n$ then if $k(x,y)$ is appropriately differentiable it satisfies a 2-parameter family of differential inequalities of which the classic triangle inequality is the order $0$ case. The generality of this result probably means it has a wealth of consequences in different contexts where reproducing kernels are relevant. As an example we point out an application to kernels of positive integral operators, which yields optimal Sobolev norm bounds.
