On the asymptotic behavior of the bestL _∞-approximation by polynomials

Abstract

LetE _n (f) denote the sup-norm-distance (with respect to the interval [−1, 1]) betweenf and the set of real polynomials of degree not exceedingn. For functions likee ^x, cosx, etc., the order ofE _n (f) asn→∞ is well known. A typical result is

$$2^{n - 1} n!E_{n - 1} (e^x ) = 1 + 1/4n + O(n^{ - 2} ).$$

It is shown in this paper that 2ⁿ⁻¹ n!E _n−1(e ^x) possesses a complete asymptotic expansion. This result is contained in the more general result that for a wide class of entire functions (containing, for example, exp(cx), coscx, and the Bessel functionsJ _k (x)) the quantity

$$2^{n - 1} n!E_{n - 1} \left( f \right)/f^{(n)} \left( 0 \right)$$

possesses a complete asymptotic expansion (providedn is always even (resp. always odd) iff is even (resp. odd)).