Non-measurable interpolation sets

Abstract

In an earlier paper the problem of the interpolation of an integral function f(z) in terms of the values taken on a sequence zn was studied by means of generalized σ-functions with zeros at zn. The purpose of the present paper is to show that the methods used in I can be modified to deal with functions of finite order and type in the angle Sα, | arg z | ≤ α. There is even a considerable saving since, for an Sα with α < π, we can escape from the awkward convergence condition Dρ (ii) of I. Most of our results, and to a great extent also their proofs, are direct generalizations of well-known results for the lattice points m + in (cf. the references at the end of I, particularly (2), (5), (6), (14), (18)).