Problems and Results on T'he Convergence and Divergence Properties of the Lagrange Interpolation Polynomials and Some Extremal Problems

Budapest In this note I will mainly discuss the joint work of Turán and mvself and some of my own results and I do not claim to give a survey of the whole subject. Let-1 c xl <. .. < xn < 1 be n points in (-1, +1). Denote bv_ l k (x) the fundamental functions of Lagrange interpolation, we have n 1, (x) = 0(x)-w(x) = L((x-xk). ~~'(xk)(x-xk~ k=1 n It is well known that the sum 1lk (x) I plays a fundamental role k=1 in the studv of the convergence and divergence properties of the Lagrauge interpolation polynomials. I proved [3], [4] sharpening previous results of Faber, Bernstein and others that for every s > 0 there is an r > 0 so that for n > n,(e,-r,) the measure of the set in x for which