Interpolation for a subclass of H ∞

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 3, August 2014, pp. 343–348.c© Indian Academy of Sciences

Interpolation for a subclass of H∞

FRANCESC TUGORES and LAIA TUGORES

Departamento de Matemáticas, Universidade de Vigo, 32004 Ourense, SpainE-mail: [email protected]; [email protected]

MS received 11 February 2013; revised 17 September 2013

Abstract. We introduce and characterize two types of interpolating sequences inthe unit disc D of the complex plane for the class of all functions being the product oftwo analytic functions in D, one bounded and another regular up to the boundary of D,concretely in the Lipschitz class, and at least one of them vanishing at some point of D̄.

Keywords. Interpolating sequence; bounded analytic function; Lipschitz class.

2000 Mathematics Subject Classification. Primary: 30E05; Secondary: 30D50.

1. Introduction

We consider the space H∞ of bounded analytic functions in D and the Lipschitz class �

of analytic functions g in D, continuous in D̄, such that

supz,w∈D

|g(z) − g(w)||z − w| < ∞.

We write (zn) for any sequence in D having no accumulation points in D. Recall thatthe pseudo-hyperbolic distance is

ψ(z,w) = |z − w||1 − z̄w| , z, w ∈ D.

For a given point zm ∈ (zn), we will choose two points in (zn), that we will denote by zm′and z∗m, verifying

zm′ ∈ {z ∈ (zn) / infi �=m

|zm − zi | = |zm − z|}

and

z∗m ∈ {z ∈ (zn) / infi �=m

ψ(zm, zi) = ψ(zm, z)}.

For a sequence (zn) satisfying the Blaschke condition

∑

n∈N

(1 − |zn|) < ∞,

343

344 Francesc Tugores and Laia Tugores

the Blaschke product with zeros at (zn) is defined as the function in H∞,

B(z) =∏

n∈N

|zn|zn

z − zn

1 − znz.

The Blaschke products with zeros at (zn)\ {zm} and (zn)\ {zm, zk} will be denoted by Bm

and Bm,k , respectively. From now on, we will put c for all positive real constants.A fundamental property of a sequence (zn) in relation to the separation of its points is

to be uniformly separated (US), that is,

|Bm(zm)| ≥ c, ∀m ∈ N (1)

and it is proved in [1] that if a sequence (zn) verifies

|Bm,m′(zm)| ≥ c, ∀m ∈ N, (2)

then it is a union of not more than two US sequences.Next we collect definitions of interpolating sequences for H∞ and �, as well as results

about them that we will need:

DEFINITION 1

(zn) is called an interpolating sequence for H∞ if given any bounded sequence ofcomplex numbers (μn), there exists f ∈ H∞ such that f (zn) = μn, ∀n ∈ N.

The well-known Carleson’s theorem [2] asserts the following:

Theorem 1. (zn) is an interpolating sequence for H∞ if and only if it is US.

DEFINITION 2

We say that (zn) is an interpolating sequence in the weak sense for H∞ if given anysequence of complex numbers (μn) verifying

|μn − μm| ≤ c ψ(zn, zm), ∀n,m ∈ N,

there exists f ∈ H∞ such that f (zn) = μn, ∀n ∈ N.

The characterization of these interpolating sequences is proved in [5].

Theorem 2. (zn) is an interpolating sequence in the weak sense for H∞ if and only if itis a union of not more than two US sequences.

Let �(zn) be the set of all functions h defined on (zn) verifying

|h(zn) − h(zm)| ≤ c |zn − zm|, ∀n,m ∈ N.

DEFINITION 3

(zn) is called an interpolating sequence for � if given any function h ∈ �(zn), there existsg ∈ � such that g(zn) = h(zn), ∀n ∈ N.

It is known that the interpolating sequences for � must be a union of not more than twoUS sequences [4].

Interpolation for a subclass of H∞ 345

We will use two results about the interpolating sequences for � when they remain in acertain region of D: the nontangential wedge, defined for a fixed t ∈ (0, 1) as

Wt ={z ∈ D

/1 − |z|2|1 − z2| > t

},

or the Stolz angle with vertex at a point η in the boundary of D and aperture β > 0,defined as

Sβ(η) ={z ∈ D

/ |z − η|1 − |z|2 < 1 + β

}.

Note that we can confine ourselves (via a rotation) to take η = 1 or η = −1. In both cases,the corresponding Stolz angle is included in the nontangential wedge for t = 1

2(1+β).

Theorem 3.

(i) If (zn) in a Stolz angle is a union of not more than two US sequences, then it is aninterpolating sequence for �.

(ii) If (zn) in a nontangential wedge is US, then it is an interpolating sequence for �.

The assertions (i) and (ii) are proved in [3] and [4], respectively. As a consequence of(i), the geometric characterization of the interpolating sequences for � when they remainin Stolz angles is to be a union of not more than two US sequences.

We denote by H∞� the class of all functions in D which are the product of one functionf in H∞ and another g in �, at least one of them vanishing at some point of D̄ (so H∞ isnot included in H∞�). It is clear that H∞� is a subclass of H∞ and therefore, in orderto introduce interpolating sequences for H∞�, we must impose some restriction to thebounded given sequence.

It is known that if f ∈ H∞, then

|f (z) − f (w)| ≤ c ψ(z,w), ∀z,w ∈ D,

and, consequently, if f ∈ H∞ vanishes on z∗m,

|f (zm)| ≤ c ψ(zm, z∗m), ∀m ∈ N. (3)

On the other hand, if g ∈ � vanishes on zm′

|g(zm)| ≤ c |zm − zm′ |, ∀m ∈ N.

Thus it is natural to pose the following interpolation problem for H∞�:

DEFINITION 4

We say that (zn) is an interpolating sequence in the weak sense for H∞� if given anysequence of complex numbers (λn) verifying

|λn| ≤ c ψ(zn, z∗n) |zn − zn′ |, ∀n ∈ N, (4)

there exists a product fg ∈ H∞�, such that

(i) f vanishes at a certain point of (zn) if and only if g vanishes at this given point;(ii) (fg)(zn) = λn, ∀n ∈ N.


Taking into account that ψ(zn, z∗n) < 1 and |zn − zn′ | < |1 − znzn′ |, ∀n ∈ N, it is

possible to pose another interpolation problem for H∞�:

DEFINITION 5

We say that (zn) is an interpolating sequence in the strong sense for H∞� if given anysequence of complex numbers (λn) verifying

|λn| ≤ c |1 − znzn′ |, ∀n ∈ N, (5)

there exists a product fg ∈ H∞�, such that

(i) if one of the two functions vanishes at a certain point of (zn), then the other functiondoes not vanish on the sequence (zn);

(ii) (fg)(zn) = λn, ∀n ∈ N.

We characterize these two types of interpolating sequences for H∞� in regions of D.Our results are as follows:

Theorem 4. (zn) in a Stolz angle is an interpolating sequence in the weak sense forH∞� if and only if it is a union of not more than two US sequences.

Theorem 5. (zn) in a nontangential wedge is an interpolating sequence in the strongsense for H∞� if and only if it is US.

We prove these theorems in the next two sections.It would be interesting to pose other interpolation problems, changing the role of �

for another space with some regularity up to the boundary of D or(and) that of H∞ fora ’nearby’ space as, for example, BMOA or the Bloch space. Also there might be lookedsome result of general type for this model of interpolating sequences, which consists inconsidering the product of two analytic functions in D with a different behaviour. Ourpaper is only a first contribution.

2. Proof of Theorem 4

Proof.

Necessity. Suppose that (zn) is an interpolating sequence in the weak sense for H∞�. For

a fixed m ∈ N, let (λn) be such that λm = zm − z∗m1 − zmz∗m

(zm − zm′) and λk = 0, if k �= m.

Since (λn) verifies (4), there exists a product fg ∈ H∞�, such that both functions f andg vanish on all points of (zn) \ {zm} and (fg)(zm) = λm.

For a non-zero function G ∈ � which vanishes on all points of (zn) \ {zm}, it is provedin [4] that

|G(z)| ≤ c |z − zk| |Bm,k(z)|, ∀k ∈ N, k �= m.

Writing this inequality for G = g, z = zm and zk = zm′ , we obtain

ψ(zm, z∗m) |zm − zm′ ||f (zm)| = |g(zm)| ≤ c |zm − zm′ | |Bm,m′(zm)|, ∀m ∈ N. (6)

Since f vanishes on z∗m, it satisfies (3), and (2) follows from (6).

Interpolation for a subclass of H∞ 347

Sufficiency. Suppose that (zn) in a Stolz angle is a union of not more than two USsequences and let (λn) be a sequence verifying (4). Let h be the function defined on (zn)

by h(zn) = zn − zn′ , ∀n ∈ N. By the triangular inequality

|h(zn) − h(zm)| ≤ |zn − zn′ | + |zm − zm′ | ≤ 2 |zn − zm|, ∀n,m ∈ N,

it follows that h ∈ �(zn). Then, by Theorem 3(i), there exists g ∈ � such that g(zn) =h(zn), ∀n ∈ N.

On the other hand, let (μn) =(

λn

zn − zn′

). By the triangular inequality and (4),

|μn−μm|≤|μn|+|μm|≤c ψ(zn, z∗n)+c ψ(zm, z∗m)≤c ψ(zn, zm), ∀n,m∈N.

Then, by Theorem 2, there exists f ∈ H∞ such that f (zn) = μn, ∀n ∈ N. The productfg ∈ H∞� performs the desired interpolation. �

3. Proof of Theorem 5

Proof.

Necessity. Suppose that (zn) in a nontangential wedge Wt is an interpolating sequence inthe strong sense for H∞�. For a fixed m ∈ N, let (λn) be such that λm = 1 − zmzm′ andλk = 0, if k �= m. We denote by � the sequence (zn) \ {zm}. Since (λn) verifies (5), thereexists a product fg ∈ H∞� such that one of the two functions vanishes on all points of�, the other does not vanish on � and (fg)(zm) = λm.

In the case where g vanishes on all points of �, we continue like in the proof of Theorem4 where the inequality in (6) becomes

|1 − zmzm′ ||f (zm)| = |g(zm)| ≤ c |zm − zm′ | |Bm,m′(zm)|, ∀m ∈ N. (7)

Since |f (zm)| ≤ c, (1) follows from (7).In the case where f vanishes on all points of �, we take into account that f = Bmu,

where u ∈ H∞ does not vanish on �, and we have

|1 − zmzm′ ||g(zm)| = |f (zm)| = |Bm(zm)| |u(zm)| ≤ c |Bm(zm)|, ∀m ∈ N. (8)

We can suppose, without loss of generality, that g vanishes on the points 1 and −1, andthen |g(zm)| ≤ c |1 − z2

m|. On the other hand, we have |1 − zmzm′ | > 12 (1 − |zm|2),

since

|1 − zmzm′ | ≥ 1 − |zm| |zm′ | > 1 − |zm| = 1 − |zm|21 + |zm| >

1

2(1 − |zm|2).

Hence, from (8) we obtain

|Bm(zm)| ≥ c1 − |zm|2|1 − z2

m|, ∀m ∈ N.

Since zm ∈ Wt , the quotient in the right-hand of this inequality is greater than t and itfollows (1).


Sufficiency. Suppose that (zn) in a nontangential wedge is a US sequence and let (λn) bea sequence verifying (5). Let h be the function defined on (zn) by h(zn) = 1 − znzn′ ,∀n ∈ N. Since

|h(zn) − h(zm)| = |zmzm′ − znzn′ | ≤ |zm| |zm′ − zn′ | + |zn′ | |zm − zn|< |zm′ − zm| + |zm − zn| + |zn − zn′ | + |zm − zn|≤ 4 |zn − zm|, ∀n,m ∈ N,

it follows that h ∈ �(zn). Then, by Theorem 3(ii), there exists g ∈ � such that g(zn) =h(zn), ∀n ∈ N.

On the other hand, let (μn) =(

λn

1 − znzn′

). By (5), we have that (μn) is a bounded

sequence. Then, by Theorem 1, there exists f ∈ H∞ such that f (zn) = μn, ∀n ∈ N. Theproduct fg ∈ H∞� performs the desired interpolation. �

References

[1] Bruna J, Nicolau A and Øyma K, A note on interpolation in the Hardy spaces of the unitdisc, Proc. Amer. Math. Soc. 124 (1996) 1197–1204

[2] Carleson L, An interpolation problem for bounded analytic functions, Amer. J. Math. 80(1958) 921–930

[3] Kotochigov A M, Free interpolation in the spaces of analytic functions with derivative oforder s from the Hardy space, J. Math. Sci. (N. Y.) 129 (2005) 4022–4039

[4] Kronstadt E P, Interpolating sequences for functions satisfying a Lipschitz condition,Pacific. J. Math. 63 (1976) 169–177

[5] Vasyunin V I, Characterization of finite unions of Carleson sets in terms of solvabilityof interpolation problems, (Russian) Investigations on linear operators and the theory offunctions, XIII. Zap. Nauchn. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 135 (1984)31–35

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Interpolation for a subclass of H ∞