A survey of published programs for best approximation

Abstract

We list and discuss published programs for best approximation of functions by linear and nonlinear families in all standard forms. In this note we list and discuss the published programs for obtaining best approximations. Let X be a set on which we wish to approximate. Most sets will be finite (an equivalent terms is discrete). Let ‖ ‖ be a norm on the continuous functions on X. Let G be a familiy of continuous functions on X. For a given basis { φ 1…, φ n}, the linear family G is the set of all functions of the form L(A,x)= ∑ k=1 n a kφ k(x) The problem of best approximation is given a continuous function f, to find g ∗ to minimize e(g) = ∥f-g∥ over gϵG. Such g ∗ is called a best approximation to f. Discrete linear approximation problems are sometimes formulated as solution of an overdetermined system of linear equations Ax = b with respect to a norm where a ij = φ j(x i) and b i = f(x) i).