Note on Degree of Approximation to Analytic Functions by Rational Functions with Preassigned Poles.

The purpose of this note is to establish several results, especially Theorem 1 below, relative to degree of approximation to analytic functions and their boundary values, the latter assumed to satisfy certain continuity conditions (Lipschitz conditions, etc.). The present results are closely related to, but more general than, results due to Sewell' and Elliott;2 the results have application to degree of approximation by bounded analytic functions. THEOREM 1. Let E be a bounded open point set of the z-plane whose boundary J consists of a finite number of mutually disjoint analytic Jordan curves Jj, J = i J,. Let f(z) be analytic on E, continuous on E + J, and of class L(p, a) on J, 0 < a < 1. In the extended plane, let.the set complementary to E + J consist of the mutually disjoint regions El, E2, . ., E,, and for each k and n let points