Approximation on a line segment by bounded analytic functions: Problem 𝛽

Problem ,B is the study of degree of approximation on a closed bounded point set E to a function f(z) analytic on E, by functions analytic and bounded in a region D containing E; here f(z) is supposed not analytic throughout D, but to possess certain continuity properties on the boundary of a suitable region of analyticity, which contains E and whose closure lies in D. This is a problem on which considerable progress has recently been made [1] if E is bounded by analytic Jordan curves. It is our present purpose to indicate further progress, now for E a line segment. We prove the