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

It 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, Korovkinʹs 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→∞.