  In this work we discuss the problem of selecting suitable approximators from
families of parameterized elementary functions that are known to be dense in a
Hilbert space of functions. We consider and analyze published procedures, both
randomized and deterministic, for selecting elements from these families that
have been shown to ensure the rate of convergence in $L_2$ norm of order
$O(1/N)$, where $N$ is the number of elements. We show that both randomized and
deterministic procedures are successful if additional information about the
families of functions to be approximated is provided. In the absence of such
additional information one may observe exponential growth of the number of
terms needed to approximate the function and/or extreme sensitivity of the
outcome of the approximation to parameters. Implications of our analysis for
applications of neural networks in modeling and control are illustrated with
examples.
