On the positivity of the fundamental polynomials for generalized Hermite-Fejer interpolation on the Chebyshev nodes

It is shown that the fundamental polynomials for (0, 1,..., 2m + 1) Hermite-Fejer interpolation on the zeros of the Chebyshev polynomials of the first kind are nonnegative for -1 less than or equal to x less than or equal to 1, thereby generalising a well-known property of the original Hermite-Fejer interpolation method. As an application of the result, Korovkin's theorem on monotone operators is used to present a new proof that the (0, 1,..., 2m + 1) Hermite-Fejer interpolation polynomials of f is an element of C[-1, 1], based on n Chebyshev nodes, converge uniformly to f as n --> infinity. (C) 1999 Academic Press.