Proof of the conjectures of Bernstein and Erdös concerning the optimal nodes for polynomial interpolation

Convergence properties of sequences of continuous functions, with kth order divided differences bounded from above or below, are studied. It is found that for such sequences, convergence in a “monotone norm” (e.g., L_p) on a, b] to a continuous function implies uniform convergence of the sequence and its derivatives up to order k ? 1 (whenever they exist), in any closed subinterval of a, b]. Uniform convergence in the closed interval a, b] follows from the boundedness from below and above of the kth order divided differences. These results are applied to the estimation of the degree of approximation in Monotone and Restricted Derivative approximation, via bounds for the same problems with only one restricted derivative.