##
Efficient Approximation of Implicitly Defined Functions: General Theorems
and Classical Benchmark Studies

#### By: Joseph W. Jerome and Anders Linner

The traditional techniques of approximation theory in the form of kernel
interpolation and cubic spline approximation are used to obtain
representations and estimates
for functions implicitly defined as solutions of two-point
boundary-value
problems. We place this benchmark analysis
in the following more general context: the
approximation of operator fixed points, not known in advance,
through a balanced combination of discretization and iteration.
We have chosen to make use of the pendulum and elastica equations, linked
by the Kirchhoff analogy, to illustrate these ideas.
In the study of these important classical models,
it is approximation theory, not numerical analysis, which is the required
theory; a significant example from micro-biology
is cited related to nucleosome repositioning.
In addition, other suggested uses of
approximation theory emerge. In particular, the determination of
approximations via symbolic calculation programs such as Mathematica is
proposed to facilitate exact error estimation.
No numerical linear inversion is required to compute the
approximations in any case.
The basic premise of the paper is that approximations should be exactly
computable in function form (up to round-off error), with error estimated
in a smooth averaged norm.
Functional analysis is employed as an effective
organizing principle to achieve this `a priori' estimation.

This paper will appear in the Journal of Approximation Theory:
145 (2007), 81--99, and
can be viewed in the following format: