On the Positivity of the Fundamental Polynomials for Generalized Hermite–Fejér Interpolation on the Chebyshev Nodes

AbstractIt is shown that the fundamental polynomials for (0,1,…,2m+1) Hermite–Fejér interpolation on the zeros of the Chebyshev polynomials of the first kind are non-negative for −1⩽x⩽1, thereby generalising a well-known property of the original Hermite–Fejér interpolation method. As an application of the result, Korovkin's 10theorem on monotone operators is used to present a new proof that the (0,1,…,2m+1) Hermite–Fejér interpolation polynomials off∈C[−1,1], based onnChebyshev nodes, converge uniformly tofasn→∞