A Note on Interpolation by Bloch Functions

We show that the positive part of a result on interpolation by Bloch functions due to B. Bøe and A. Nicolau is a direct consequence of C. Sundberg’s description of the traces of BMOA functions on interpolating sequences for H∞. The Bloch space B consists of all the analytic functions f in the unit disk D of the complex plane such that ‖f‖B = sup z∈D (1− |z|)|f ′(z)| < ∞. Recall that the hyperbolic distance between two points z, w ∈ D is d(z, w) = 1 2 log 1 + ρ(z, w) 1− ρ(z, w) , where ρ(z, w) = |(z−w)/(1− zw)| is the pseudohyperbolic distance between them. It turns out that an analytic function f in D belongs to B if and only if it is a Lipschitz function from D to C, when D is equipped with the hyperbolic metric and C with the Euclidean metric. Indeed, ‖f‖B = sup{|f(z)− f(w)|/d(z, w) : z, w ∈ D, z = w}. Thus it seems natural to say that a sequence of points {zj}j≥1 in D is interpolating for B if for any sequence of complex numbers {αj}j≥1 satisfying the Lipschitz condition |αj − αk| = O(d(zj , zk)) there is some f ∈ B such that f(zj) = αj , for every j ≥ 1. B. Bøe and A. Nicolau in [1] obtained a very nice geometrical characterization of those sequences by means of the following result. Theorem A. A sequence {zj}j≥1 in D is interpolating for B if and only if it is the union of at most two separated sequences and satisfies the following condition: (A) There exists a constant 0 < α < 1 such that #{ zj : ρ(zj , z) < r } (1− r)−α (z ∈ D, 0 < r < 1). Recall that a sequence {zj}j≥1 in D is separated when infj =k ρ(zj , zk) > 0. As usual, the notation A B means that A is less than or equal to a constant times Received by the editors November 30, 2005 and, in revised form, March 16, 2006. 2000 Mathematics Subject Classification. Primary 30E05, 30D45; Secondary 30D50.