POLYNOMIAL-APPROXIMATION ON THE BOUNDARY AND STRICTLY INSIDE

We investigate the possibility of approximating a function on a compact set K of the complex plane in such a way that the rate of approximation is almost optimal on K, and the rate inside the interior of K is faster than on the whole of K. We show that if K has an external angle smaller than pi at some point z0 is-an-element-of partial derivative K, then geometric convergence inside K is possible only for functions that are analytic at z0. We also consider the possibility of approximation rates of the form exp(-cn(beta)) for approximation inside K, where beta is related to the largest external angle of K. It is also shown that no matter how slowly the sequence {gamma(n)} tends to zero, there is a K and a Lip beta, beta < 1, function f such that approximation inside K cannot have order {gamma(n)}.