On the fundamental polynomials for Hermite-Fejer interpolation of Lagrange type on the Chebyshev nodes

For a fixed integer m greater than or equal to 0 and 1 less than or equal to k less than or equal to n, let A(k,2m,n)(T,x) denote the kth fundamental polynomial for (0,1,...,2m) Hermite-Fejer interpolation on the Chebyshev nodes {x(j,n) = cos[(2j - 1)pi/(2n)]: 1 less than or equal to j less than or equal to n}. (So A(k,2m,n)(T, x) is the unique polynomial of degree at most (2m + 1)n - 1 which satisfies A(k,2m,n) (T, x(j,n)) = delta(k,j), and whose first 2m derivatives vanish at each x(j,n).) In this paper it is established that \A(k,2m,n)(T,X)\ less than or equal to A(1,2m,n)(T,1), 1 less than or equal to k less than or equal to n, -1 less than or equal to x less than or equal to 1. It is also shown that A(1,2m,n)(T, 1) is an increasing function of n, and the best possible bound C-m so that \A(k,2m,n)(T, x)\ < C-m for all k, n and x is an element of [-1, 1] is obtained. The results generalise those for Lagrange interpolation, obtained by P. Erdos and G. Grunwald in 1938.