Asymptotic expansion for the approximation of the Lipschitz function by the Hermite-Fejér interpolation polynomial (Q1347157)

The Hermite-Fejér interpolation polynomial associated with \(f\in C[-1, 1]\) and based on \(x_ k= \cos (2k-1) \pi/ (2n)\), \(k=1, \dots, n\), can be denoted by \[ H_{2n-1} (f,x)= n^{-2} T_ n^ 2 (x) \sum_{k=1}^ n f(x_ k) (1-xx_ k) (x- x_ k^{-2}), \] where \(T_ n (x)\) are the Chebyshev polynomials. The paper gives the pointwise complete asymptotic expansion for \(\Delta_ n (x)= \sup_{f\in \text{Lip }1} | H_{2n-1} (f,x)- f(x) |\), where \(\text{Lip }1\) is the Lipschitz class \(\text{Lip } 1= \{f\in C[-1, 1]\): \(| f(x)- f(y)|\leq | x-y|\), \(x,y\in [-1,1]\}\).