  A function approximation method is developed that aims to approximate a
function in a small neighborhood of a state that travels within a compact set.
The development is based on the theory of universal reproducing kernel Hilbert
spaces over the $n$-dimensional Euclidean space. Several theorems are
introduced that support the development of this State Following (StaF) method.
In particular, it is shown that there is a bound on the number of kernel
functions required for the maintenance of an accurate function approximation as
a state moves through a compact set. Additionally, a weight update law, based
on gradient descent, is introduced where arbitrarily close accuracy can be
achieved provided the weight update law is iterated at a sufficient frequency,
as detailed in Theorem 6.1.
  To illustrate the advantage, the impact of the StaF method is that for some
applications the number of basis functions can be reduced. The StaF method is
applied to an adaptive dynamic programming (ADP) application to demonstrate
that stability is maintained with a reduced number of basis functions.
  Simulation results demonstrate the utility of the StaF methodology for the
maintenance of accurate function approximation as well as solving an infinite
horizon optimal regulation problem through ADP. The results of the simulation
indicate that fewer basis functions are required to guarantee stability and
approximate optimality than are required when a global approximation approach
is used.
