THE CLOSED CONVEX HULL OF THE INTERPOLATING BLASCHKE PRODUCTS

The closed convex hull of the interpolating Blaschke products contains any bounded analytic function of sufficiently small norm. Let H∞ be the set of bounded analytic functions on the open unit disc D. Each f ∈ H∞ has a non-tangential limit f(e) at almost every point e ∈ T = ∂D. An inner function f is a bounded analytic function satisfying |f(eiθ)| = 1, a.e. e ∈ T . If {zn} ⊂ D satisfies ∑ (1 − |zn|) < ∞, then ∞ ∏ n=1 −zn |zn| z − zn 1 − znz is called the Blaschke product with zeros {zn}. If zn = 0, replace −zn |zn| by 1. A Blaschke product is an inner function. These products have a lot of interesting properties: If f ∈ H∞ then there is a Blaschke product B and a non-vanishing g ∈ H∞ such that f = Bg. If I is an inner function then I − λ I − λI is a Blaschke product for λ in a dense subset of D. Therefore the Blaschke products are dense in the inner functions in the uniform norm. The closed convex hull of the Blaschke products is the unit ball of H∞. See [G] for the proof of these and many other properties of the Blaschke products. A square is a set of the form Q = {re : 1 − h < r < 1, θ0 < θ ≤ θ0 + h}.

is called the Blaschke product with zeros {z n }.If z n = 0, replace −zn |zn| by 1.A Blaschke product is an inner function.These products have a lot of interesting properties: If f ∈ H ∞ then there is a Blaschke product B and a non-vanishing g ∈ H ∞ such that f = Bg.If I is an inner function then I − λ I − λI is a Blaschke product for λ in a dense subset of D. Therefore the Blaschke products are dense in the inner functions in the uniform norm.The closed convex hull of the Blaschke products is the unit ball of H ∞ .See [G] for the proof of these and many other properties of the Blaschke products.
A square is a set of the form The length of Q is |Q| = h.The pseudohyperbolic metric on D is defined by and for all squares.
Given an square Q, its top half is A Blaschke product whose zero set is an interpolating sequence is called an interpolating Blaschke product.These products have been studied in great detail, and they play a significant role in the theory of H ∞ , see [G].An important open problem is whether or not they are dense in the set of inner functions.In this paper we study the closed convex hull K of the interpolating Blaschke products.
The proof of this modest result is long and technical.The ideas may disappear in the formalism, so I delete details at some points.
Let B(z) be a Blaschke product with zeros {z n }.Assume that ρ(z, z n ) is close to 1 for all n.Then Assume that the zeros are contained in disjoint squares Assume that the density of all Q k is smaller than a number d.Then Here z = re iθ .This proves Lemma 0. Under the above assumptions, |B(z)| ≥ e −πd(1+0(1)) .
Lemma 1.Given a square Q, and constants Below we consider only the dyadic squares: where a is close to 1 and a N = δ.The zeros are Let R ⊂ Q be the region bounded by Γ = {z ∈ Q : |z| = a M } and a circle intersecting Γ in a small angle γ, as in Figure 1.
R is bounded by Γ and another circle.The angle of intersection, γ, is small.The corners of R are far, but not too far, from the endpoints of Γ.
The figure does not tell the truth: Most of Q ∩ {z : |z| < a} is contained in R, that is: All points in Q ∩ {|z| < a} are "close" to R in the pseudohyperbolic metric.The argument proving Lemma 0 shows that for z ∈ R the contribution to |B a,N (z)| coming from the zeros outside Q is close to 1, provided the corners of R are moved away from the endpoints of Γ.By making γ small and increasing the distance from the corners of R to the endpoints of Γ, we obtain By increasing M , decreasing κ slightly and making γ small, we finish the proof.
Lemma 3. Let B(z) be a B-product with zeros {z n }.Assume that there exists constants A < 1, η > 0 such that for all n there exists z By Lemmas 4 and 2, we may assume that the conclusion of Lemma 2 holds.
Given δ > 0 close to 1. Let 0 < δ < β < 1 to be chosen later.Choose . .be the maximal dyadic squares such that 1. inf These are the squares of the first generation.The proof of Lemma 0 shows that The second generation are the maximal dyadic squares such that 1. inf (See [G, p. 332].) n }, where M , Γ k i,n and R k i,n come from Lemma 1 and its proof.See figure below.
The estimate (1) shows . By choosing δ close to 1 we obtain That is: We have control over the density of Choose r < κ; r slightly smaller.Then r is close to 1.By Frostman's theorem we can choose r such that where B Q takes care of the zeros inside Q w (see Figure 3), and consider the following three different situations.
Case 1: ρ(w, z n ) is large for all n and every S k 1,n meeting Q w is small, say, . The world looks like this: Let us consider all the disjoint dyadic subsquares of Q w , of length 100η(1 − |w|).Then the density of all these small, but not too small, subsquares of Q w is less than ). Lemma 3 does the trick.
Case 2: ρ(w, z N ) is small for some N .
Here small means not close to 1. Then z N plays the same role as w * in Case 1.
Remark.This argument will be repeated later, but we will not repeat the details.The parameter of the Frostman transform, δ r , satisfied two conditions: 1 2. The density d of Q k n w.r.t.B 1 B * 1 satisfied e −πd > δ κ .Therefore the argument with a Frostman parameter δ * works if It is easy to see that the unimodular constants belong to the closed convex hull K of the interpolating B-products.We have Since δ > 0, a computation proves that Alternatively.Repeat the argument with δ r replaced by δ r e iθ .For almost all θ the function Approximating with Riemann sums we see that This leads to B 1 B * 1 ∈ 1 1−δ 2r K.This estimate is better than the previous one if δ r is small.
We now want to repeat the argument with B * 1 replaced by

Lemma 4 .
Any finite product of interpolating B-products can be uniformly approximated by an interpolating B-product.Proof: See [M-S].A consequence of Lemma 4 is that the closed convex hull of the interpolating B-products is closed under multiplication.
there exist a finite Blaschke product B with zeros on (|z| = a) ∩ Q and a region R of the unit disc, such that 1.The zeros of B are uniformly distributed on |z| = a 2. The density of Q w