On the asymptotic behavior of the best \(L_{\infty}\)-approximation by polynomials (Q1096826)

Let \(E_ n(f)\) denote the Chebyshev distance (with respect to the interval [-1,1]) between f and the set of polynomials of degree not exceeding n. This paper proves that for a wide class of entire functions (containing exp(cx), cos(cx) and the Bessel functions \(J_ k(x))\) the quantity \(2^{n-1}n!E_{n-1}(f)/f^{(n)}(0)\) possesses a complete asymptotic expansion (provided n is always even (resp. always odd) if f is even (resp. odd)).