On the Uniform and Simultaneous Approximations of Functions

Abstract

We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F 1 and F 2 are continuous functions on a closed interval [ a , b ], S is an n -dimensional Chebyshev subspace of C [ a , b ] and s 1 * & s 2 * are the best uniform approximations to F 1 and F 2 from S respectively. The characterization of the best approximation solution is used to show that, under some restrictions on the point set of alternations of F 1 − s 1 * and F 2 − s 2 * , s 1 * or s 2 * is also a best A (1) simultaneous approximation to F 1 and F 2 from S with F 1 ≥ F 2 and n =2.