On the uniform convergence of interpolating polynomials

摘要

A sufficient condition is given for a continuous power series f(x) on [0, 1] to be the uniform limit of its sequence Pnf of interpolating polynomials at n + 1 equally spaced nodes. The proof is based on expanding the Newton coefficients of Pnf in terms of Stirling numbers of the second kind and applying an Abel-like summation formula. Convergence rates of Pnf and of related coefficient sequences are estimated. Similar results follow for Bernstein polynomials and their derivatives.