The complete asymptotic expansion for the degree of approximation of Lipschitz functions by Hermite-Fejér interpolation polynomials (Q1061332)

This article is concerned with the asymptotic behaviour as \(n\to \infty\) of the maximum interpolation error: \(\Delta_ n:=\sup \{\| H_{2n- 1}(f)-f\|_{\infty}: f\in Lip 1\}\). Here \(H_{2n-1}(f;x)\) is the Hermite-Fejér interpolation polynomial of degree not exceeding 2n-1 associated with f and based on the zeros of the Chebyshev polynomial of the first kind, \(T_ n(x)\), and Lip 1 is the class of all Lipschitz 1 functions on [-1,1]. It is shown that \[ \Delta_ n\sim (2/\pi)(\log n/n)+(2/\pi)[\log (8/\pi)+\gamma]1/n+4\sum^{\infty}_{k=1}(A^*_ k/(2k+1))\quad as\quad n=2m\to \infty, \] where \(A^*_ k=(-1)^{k- 1}(2^{2k-1}-1)^ 2B^ 2_{2k}/(2k)(2k)!(\pi /2)^{2k-1}\) for \(k=1,2,3,...\), \(\gamma\) is Euler's constant and \(B_{2k}\) for \(k=1,2,3,...\), represent Bernoulli numbers.