A Comparison of ``Best'' Polynomial Approximations with Truncated Chebyshev Series Expansions
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Introduction. In the numerical solution of mathematical problems it is common to represent a function of a real variable by the leading terms of its infinite Chebyshev series expansion. The purpose of this paper is to compare the accuracy of such a polynomial approximation with that of the "best" polynomial approximation of the same degree (the "best" being the one whose greatest error is as small as possible). We shall refer to these approximations as the CS and BA respectively; they are defined and discussed in [1]. By definition the BA is more accurate than the CS; on the other hand, the CS is easier to compute. It is practical, therefore, to enquire whether in any given situation the increase in accuracy is sufficient to justify the extra labour of computing the BA. This paper aims to provide some guidance on this question. (We may note that the arguments for and against expressing a polynomial approximation in terms of powers of the independent variable rather than in terms of Chebyshev polynomials are irrelevant here; the BA and the CS can, of course, both be written in either form.) Our procedure is to calculate the maximum absolute errors of the BA and the CS of degree n (which errors we shall denote by En and Sn re? spectively, following Bivlin [2]) for an arbitrary polynomial of degree n + r + 1, with r = 0, 1, 2 and 3. In each case we compute R(r), the maxi? mum value of Sn/En over all polynomials of degree n + r + 1. From these results we conjecture the value of R(r) for arbitrary r, and support this conjecture by numerical experiment in the case r ? 5. The method of attack is based on a device used by Bernstein [4] for a closely related prob? lem. Some related work on the comparison of the CS with the BA has been carried out by Rivlin [2]. He presents some interesting evidence which, as he says, "seems to support the practice of truncating the Chebyshev series as an approximation to the best approximation." Using some theorems proved by Blum and Curtis [3], he gives bounds for Sn/En which are most useful in the case of rapidly convergent Chebyshev series. Here, however, we shall consider the problem under less restrictive conditions.