H-Sets for Non-Linear Constrained Approximation

Abstract

The idea of an H-set was first proposed by Collatz [4], which was primarily introduced to give lower bounds for the numerical calculation of best linear approximation by sets not satisfying the Haar condition. The usefulness of H-sets for the characterisation and anlysis of best approximation in this non-Haar setting is set out in Brannigan [1]. A complete exposition in terms of functions in a normed linear setting is given in Brannigan [2]. For non-linear approximation an extension of H-sets is given in Collatz and Krales [5]. We here consider the non-linear approximation problem where constraints on the parameter set are present. This setting is of most practical use, and the analysis of numerical non-linear constrained approximation is needed. For the general characterisation of best approximation see Brannigan [3]. The usefulness of H-sets as developed here is in the ability to compute such sets in many cases. In the following the theorems are merely stated, those interested should consult Brannigan [3].