The Chebyshev solution of an overdetermined system of linear equation is considered when linear constraints on the solution are present. Constraints considered are constraints on the given matrix of coefficients plus general linear constraints. No assumption on the matrix of coefficients, such as the Haar condition, is made; thus the most general discrete linear problem is considered. Use is made of the concept of H-sets to give characterization theorems. An ascent exchange algorithm is given with numerical examples. Convergence of the algorithm when the problem has a solution is proven, and a computable condition is given for the case when the constraints make the problem infeasible.

Discrete Chebyshev Approximation with Linear Constraints

The theory of H-sets as originally defined by L. Collatz has been developed in recent papers and shown to be a unifying concept for linear approximation problems. We here extend the theory of H-sets for non-linear uniform approximation with functional constraints where the approximating functions have a compact domain and range in a Banach space. With the theory of H-sets as here developed the characterization of global and local best approximations follow, and conditions for local to be global approximations are given. A convergence theorem for a descent algorithm is given.

Uniform non-linear constrained approximation in normed linear spaces

A geometric characterization of H-sets

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].

H-Sets for Non-Linear Constrained Approximation

Approximation by semi-non-linear functions

It has been shown that the theory of H-sets is important in the characterization of best uniform approximation of continuous real-or complex-valued functions. We here extend the theory of H-sets to the more general setting of functions with compact domain and with range contained in a Banach space. Using the definitions of H-sets, we construct a maximal linear functional and obtain inclusion theorems analogous to the classical case. It is then a simple matter to deduce a characterization of best approximation and show when uniqueness and strong uniqueness are achieved.

M. Brannigan

Papers

Uniform linear approximation in normed linear spaces