  In this paper, the evolution equation that defines the online critic for the
approximation of the optimal value function is cast in a general class of
reproducing kernel Hilbert spaces (RKHSs). Exploiting some core tools of RKHS
theory, this formulation allows deriving explicit bounds on the performance of
the critic in terms of the kernel and definition of the RKHS, the number of
basis functions, and the location of centers used to define scattered bases.
The performance of the critic is precisely measured in terms of the power
function of the scattered basis used in approximations, and it can be used
either in an a priori evaluation of potential bases or in an a posteriori
assessments of value function error for basis enrichment or pruning. The most
concise bounds in the paper describe explicitly how the critic performance
depends on the placement of centers, as measured by their fill distance in a
subset that contains the trajectory of the critic.
