A Comparison of Uniform Approximations on an Interval and a Finite Subset Thereof

[. xEX If a finite-dimensional linear subspace M is prescribed in C[X] and if f is any element of C[X], then there exists inM an element of best approximation to f. In other words, the distance from f to the setM is achieved by at least one element of M. The chief example of such approximation is that in which X is an interval and M is taken to be the space of polynomials having degree -<n. In all but certain exceptional eases, the determination of a best approximation is a complicated numerieM process which involves the solution of systems of nonlinear equations. Because of this difficulty, various approximate methods are often substituted. One of the simplest of these consists in selecting a finite subset Y inX and then solving the approximation problem on Y instead of X. The latter task can be accomplished by simpler procedures requiring only a finite number of steps. The reliance which may be placed upon the resulting approximation as a substitute for the best approximation depends upon a number of factors, especially the of the subset Y and the modulus of continuity of f. These matters were first studied by Motzkin and Walsh in [7], and that work has inspired the present investigation. Our results below are arranged so that many of the general theorems are followed by one or more results specialized to the ease of algebraic polynomials on the interval [-1, 1]. The specialized results are labeled with suffixes a, b, c, etc. Some of the theorems given here appeared several years ago in a research report of limited distribution [3]. 2. If Y is any subset of X, we define the density of Y to be the number