Efficient approximation of implicitly defined functions: General theorems and classical benchmark studies

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.