On Interpolation and Approximation by Functions Analytic and Bounded in a Given Region.

The general problem of interpolation and approximation to an arbitrary given analytic function by means of polynomials or more general rational functions, is an important one on which the existing literature is large, including a recent book by the present writer. ' Analogous to this problem is the new but related problem of interpolation and approximation to an arbitrary given analytic function by means of functions required to be analytic and to have a given bound in a given region. The new problem is perhaps more difficult than the old, but nevertheless some of the technique that has already been developed for the former problem applies with relatively minor modification to the study of the new problem. It is the object of the present note to indicate briefly some of these modifications and their consequences. The present note represents, however, only the beginnings of the study of the new problem, and many open questions are left untouched. I.-PROBLEM I. Let the closed point set S lie interior to the region R, and let the function f(z) be analytic on S but let f(z) (considered together with its possible analytic extensions) not be analytic throughout the interior of R. Given M > 0, to study the function or all functions Fm(z) analytic and of modulus not greater than M in R, such that the quantity