Pointwise estimates for lagrange interpolation polynomials

Let f is an element of C[-1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes {cos 2k-1/2n pi} boolean OR {-1, 1} be Delta(n+2)(f, x). In this paper we study the estimate of Delta(n+2)(f, x), that keeps the interpolation property. As a result we prove that Delta(n+2)(f, x) = Omicron(1) {omega(f, root 1-x(2)/n)vertical bar T-n(x)vertical bar ln (n + 1) + omega(f, root 1-x(2)/n vertical bar T-n(x)vertical bar)}, where T-n(x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f is an element of C-r[-1, 1] with r >= 1, then Delta(n+2)(f, x) = Omicron(1) {root 1-x(2)/n(r)vertical bar T-n(x)vertical bar omega(f((r)),root 1-x(2)/n) ((root 1-x(2) + 1/n)(r-1) ln(n + 1) + 1)}.