Finite Unions of Interpolation Sequences

A unified and relatively simple proof is given for some well-known results involving finite unions of uniformly separated sequences. In this note we consider finite unions of uniformly separated sequences, which appear in several well-known theorems involving both Hardy and Bergman spaces. The techniques of proof, some of which are quite involved, differ greatly from result to result. The purpose of this note is to provide a relatively simple and selfcontained proof that unifies all of these results. A sequence {zk} of points in the unit disk D = {z ∈ C : |z| < 1} is called a Blaschke sequence if ∞ ∑ k=1 (1− |zk|) < ∞. It is uniformly separated if ∏ j 6=k ∣∣∣∣ zj − zk 1− zjzk ∣∣∣∣ ≥ δ , k = 1, 2, . . . , for some constant δ > 0 that is independent of k. The importance of uniformly separated sequences is evident in their role as interpolation sequences for the Hardy space H, the set of functions f analytic in the disk satisfying ‖f‖Hp = sup r<1 { 1 2π ∫ 2π 0 |f(reiθ)|pdθ }1/p < ∞. A sequence {zk} in the disk is said to be a universal interpolation sequence if for each complex sequence {wk} ∈ `∞ there exists a bounded analytic function f with f(zk) = wk for k = 1, 2, . . . . Carleson [1] proved that {zk} is a universal interpolation sequence if and only if it is uniformly separated. Shapiro and Shields [13] then generalized the theorem to arbitrary H spaces (1 ≤ p < ∞) by showing that the operator Tp(f) = { (1− |zk|)f(zk) } 1991 Mathematics Subject Classification. Primary 30H05, 46E15.