Approximation by φ( ax) L( A, x) on finite point sets

Abstract

Let X − { x 1,…, x N } be a finite subset of the real line, x 1− … x N . Let φ be a continuous function on the real line and { ψ 1,…, ψ n } a Chebyshev set on X, n < N. Define L( A, x) − ∑ k−1 n a k ψ k ( x), F( A, x) − φ( a 0 x) L( A, x). Let φ be a given norm on the functions on X. Let G be a family of functions containing { F( A, ·)}. The approximation problem is: Given a function f on X, find g ∗ ϵ G for which ∥ f … g∥ attains its infimum ϱ( f) over g ϵ G. Such an element g ∗ is called a best approximation. In this note we consider the existence of best approximations.