Interpolation problems in function spaces

Abstract

Let D be a domain in the complex plane, let {zn} be a sequence of distinct points in D, and let {wn} be an arbitrary sequence of complex numbers. Given a space E of functions on D, the problem arises to characterize the pairs of sequences {zn} and {wn} for which there is a function [function of (italic small f)]; [epsilon] E with [function of (italic small f)];(zn) = wn, n = 1, 2,.... In the present paper, we solve a general interpolation problem of this type. We then apply the result to obtain criteria for interpolation by Hp functions, 1 [les] p [les] [infinity], by harmonic functions of class hp, and by functions belonging to certain Hilbert spaces. The main tool is a general theorem, closely related to the Hahn-Banach theorem, on the extension of functionals over normed linear spaces.