State following (StaF) kernel functions for function approximation Part I: Theory and motivation

Unlike traditional methods that aim to approximate a function over a large compact set, a function approximation method is developed in this paper that aims to approximate a function in a small neighborhood of a state that travels within a compact set. The development is based on universal reproducing kernel Hilbert spaces over the n-dimensional Euclidean space. Three theorems are introduced that support the development of this state following (StaF) method. In particular an explicit uniform number of StaF kernel functions can be calculated to ensure good approximation as a state moves through a large compact set. An algorithm for gradient descent is demonstrated where a good approximation of a function can be achieved provided that the algorithm is applied with a high enough frequency.