On Some Convergence Properties of the Interpolation Polynomials

Abstract

It is well known that there exist continuous functions whose Lagrange interpolation polynomials taken at the roots of the Tchebycheff polynomials T„ (x) diverge everywhere in (-1, + 1) .' On the other hand a few years ago S . Bernstein proved the following result' : Let f(x) be any continuous function ; then to every c > 0 there exists a sequence of polynomials ~p„(x) where ~0 ,(x) is of degree n 1 and it coincides with f(x) at, at least n cn roots of T, (x) and gyp, (x) -p f( .c) uniformly in (-1, + 1) . Fejer proved the following theorem' : Let the fundamental points of the interpolation be a normal 4 point group