On the uniform convergence of interpolating polynomials (Q1015775)

Let \(f:[0,1] \to \mathbb R\) be a continuous function defined by a power series, and let \(P_n\) be the \(n\)th degree polynomial that interpolates \(f\) at \(n+1\) equispaced points. The paper states sufficient conditions for the sequence \((P_n)\) to converge uniformly against \(f\). Error estimates for this procedure and for the approximation of \(f\) by Bernstein polynomials are provided, too. The proofs avoid methods using complex analysis.