An Interpolation Problem for Bounded Analytic Functions

Abstract

be possible for a given sequence of points {zv}, I z, 1, and an analytic function f(z) in I z i < 1, I f(z) I _ 1. The result is, however, very implicit ancd gives in a concrete situation very little help in deciding if the interpolatioln is possible or not. The object of the present paper is to give a simple and explicit condition on {z,} under which the interpolation (1. 1) is possible with a bounded function f(z). If we allow {w,} to be an arbitrary bounded sequence, the condition is also a necessary one. It should be observed that even the existence of any infinite such sequence {zv} is non-trivial; this problem was suggested by R. C. Buck and constructions of such examples have also recentlybeen given by G-leason and Newman (unpublished). The proof of the main theorem depends on a reformulation of problem (1.1) which is presented in section 2. It is essentially included in a result by Garabedian [2]; since the discussion there is quite general and the proof complicated, we have included a complete and simple proof here. Section 3 contains an inequality of the Schwarz type, which is the crucial step in our proof. This is then completed in section 4. The last section is devoted to an application to the ideal structure in the algebra of bounded analytic functions.