Extremely weak interpolation in $H^{\infty}$

Given a sequence of points in the unit disk, a well known result due to Carleson states that if given any point of the sequence it is possible to interpolate the value one in that point and zero in all the other points of the sequence, with uniform control of the norm in the Hardy space of bounded analytic functions on the disk, then the sequence is an interpolating sequence (i.e.\ every bounded sequence of values can be interpolated by functions in the Hardy space). It turns out that such a result holds in other spaces. In this short note we would like to show that for a given sequence it is sufficient to find just {\bf one} function interpolating suitably zeros and ones to deduce interpolation in the Hardy space.


INTRODUCTION
The Hardy space H ∞ of bounded analytic functions on D is equipped with the usual norm f ∞ = sup z∈D |f (z)|. A sequence Λ = {λ n } n ⊂ D of points in the unit disk is called interpolating for H ∞ , noted Λ ∈ Int H ∞ , if every bounded sequence of values v = (v n ) n ∈ l ∞ can be interpolated by a function in H ∞ . Clearly, for f ∈ H ∞ , the sequence (f (λ n )) n is bounded. Hence Λ ∈ Int H ∞ def ⇐⇒ H ∞ |Λ = l ∞ (we identify the trace space with a sequence space). The sequence Λ is said to satisfy the Blaschke condition if n (1 − |λ n |) < ∞. In that case, the Blaschke product B = n b λn , where b λ (z) = |λ| λ z−λ 1−λz is the normalized Möbius transform (λ ∈ D), converges uniformly on every compact set of D to a function in H ∞ with boundardy values |B| = 1 a.e. on T. Carleson proved (see [Ca58]) that The latter condition will be termed Carleson condition, and we shall write Λ ∈ (C) when Λ satisfies this condition. Carleson's result can be reformulated using the notion of weak interpolation. Definition 1.1. A sequence Λ of points in D is called a weak interpolating sequence in H ∞ , noted Λ ∈ Int w H ∞ , if for every n ∈ N there exists a function ϕ n ∈ H ∞ such that • for every n ∈ N, ϕ n (λ k ) = δ nk , • sup n ϕ n ∞ < ∞.
Date: October 19, 2010. 1991 Mathematics Subject Classification. 30E05, 32A35. Key words and phrases. Hardy spaces, interpolating sequences, weak interpolation. This project was elaborated while the author was Gaines Visiting Chair at the University of Richmond, also partially supported by the french ANR-project FRAB. Now, when Λ ∈ (C), setting ϕ n = B n /B n (λ n ) we obtain a family of functions satisfying the conditions of the definition. Hence Λ ∈ Int w H ∞ . And from Carleson's theorem we get With suitable definitions of interpolating and weak interpolating sequences, such a result has been shown to be true in Hardy spaces H p (see [ShHSh] for 1 ≤ p < ∞ and [Ka63] for 0 < p < 1) as well as in Bergman spaces (see [SchS98]) and in certain Paley-Wiener and Fock spaces (see [SchS00]).
One also encounters the notion of "dual boundedness" for such sequences (see [Am08]), and in a suitable context it is related to so-called uniform minimality of sequences of reproducing kernels (see e.g. [Nik02, Chapter C3] for some general facts).
Using a theorem by Hoffmann we want to show here that given a separated sequence Λ, then there is a splitting of Λ = Λ 0 ∪ Λ 1 such that if there exists just one function f ∈ H ∞ vanishing on Λ 0 and being 1 on Λ 1 , then the sequence is interpolating in H ∞ .
The author does not claim that such a result is anyhow useful to test whether a sequence is interpolating or not, but that it might be of some theoretical interest.
In view of this theorem, given any Blaschke sequence of points Λ in the disk and a constant δ ∈ (0, 1), we will set Λ 0 to be the zero set of B 0 and Λ 1 to be the zero set of B 1 where B = B 0 B 1 is a Hoffman factorization of B. We will refer to Λ = Λ 0 ∪ Λ 1 as a δ-Hoffman decomposition of Λ.
Replacing f by g = 1 − f we obtain a function vanishing now on Λ 1 and being 1 on Λ 0 . And the same argument as before shows that for µ ∈ Λ 0 |B 1 (µ)| ≥ η (let us agree to use the same η here).
We have to check whether such an estimate holds also for the second piece. Now, let z ∈ ∂D(µ, δ) (note that δ = δ 0 /2, where δ 0 is the separation constant of Λ, so that this disk is far from the other points of Λ). Then by Hoffman's theorem Now B Λ 1 \{µ} and B 0 do not vanish in D(µ, δ). We thus can take powers of B 0 and divide through getting a function B Λ 1 \{µ} /B 1/b 0 not vanishing in D(µ, δ). By the minimum modulus principle we obtain B Λ 1 \{µ} (z) for every z ∈ D(µ, δ) and especially in z = µ so that By the preliminary observation above, the same argument can be carried through when µ ∈ Λ 0 , so that for every µ ∈ Λ we get |B Λ\{µ} (µ)| ≥ c for some suitable c > 0. Hence Λ ∈ (C) and we are done.

Remark 2.3. 1) It is clear from the proof that it is sufficient that there is an
This means that in terms of Blaschke products, we need two functions instead of the sole function f (which is of course not unique) as stated in the theorem. One could raise the question whether it would be sufficient to have only one of the conditions in (2.1) (the condition is clearly necessary). Suppose we had the first condition inf µ∈Λ 1 |B 0 (µ)| ≥ η Then in order to obtain the condition of the theorem, we would need to multiply B 0 by a function F ∈ H ∞ such that (B 0 F )(µ) = 1 for every µ ∈ Λ 1 . In other words we need that B 0 + B 1 H ∞ is invertible in the quotient algebra H ∞ /B 1 H ∞ under the condition that 0 < η ≤ |B 0 (µ)| ≤ 1. This is possible when Λ 1 is a finite union of interpolating sequences in H ∞ (which in our case boils down to interpolating sequences since we have somewhere assumed that Λ, and hence Λ 1 , is separated). See for example [Har96] for this, but it can also be deduced from Vasyunin's earlier characterization of the trace of H ∞ on finite union of interpolating sequences (see [Vas84]).
We do not know the general answer to this invertibility problem when Λ 1 is not assumed to be a finite union of interpolating sequences.
2) Another question that could be raised is whether in Theorem 2.2 the assumption of being separated can be abandoned. At least Hoffman's theorem does not allow us to deduce that the sequence is separated. As an example, one could have a union of two interpolating sequence the elements of which come arbitrarily close to each other. Write Λ = n σ n where σ n contains two close points of Λ one of which is of the first interpolating sequence and the other one from the second interpolating sequence. Let Λ 0 be the union of the even indexed σ n 's and Λ 1 the odd indexed σ n 's we obtain a Hoffman decomposition for which we can find f as in the theorem, but Λ is not interpolating.