Reducing sequences
We introduce and examine two new classes of distinguished sequences in the unit disk for the space of bounded analytic functions. One of these sequences is intermediate between interpolating and zero sequences.
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| Datum: | 2017 |
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Institute of Mathematics, NAS of Ukraine
2017
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507530896080896 |
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| author | Tugores, F. Тугорес, Ф. |
| author_facet | Tugores, F. Тугорес, Ф. |
| author_sort | Tugores, F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2019-12-05T09:23:56Z |
| description | We introduce and examine two new classes of distinguished sequences in the unit disk for the space of bounded analytic
functions. One of these sequences is intermediate between interpolating and zero sequences. |
| first_indexed | 2026-03-24T02:10:47Z |
| format | Article |
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UDC 517.5
F. Tugores (Univ. Vigo, Spain)
REDUCING SEQUENCES
ЗВIДНI ПОСЛIДОВНОСТI
We introduce and examine two new classes of distinguished sequences in the unit disk for the space of bounded analytic
functions. One of these sequences is intermediate between interpolating and zero sequences.
Введено та вивчено два нових класи видiлених послiдовностей в одиничному колi для простору обмежених аналi-
тичних функцiй. Одна з цих послiдовностей є промiжною мiж iнтерполяцiйною та нульовою послiдовностями.
1. Introduction. Let \BbbD be the unit disk of the complex plane. We consider the space H\infty of all
analytic functions f in \BbbD such that \| f\| \infty = \mathrm{s}\mathrm{u}\mathrm{p}z\in \BbbD | f(z)| < \infty and the Bergman space A1 of all
analytic functions in \BbbD such that
\| f\| 1 =
\int
\BbbD
| f(z)| dm(z) < \infty .
Let l1 and l\infty be the Banach spaces of all complex sequences (an) such that \| (an)\| 1 =
\sum \infty
n=1
| an| <
< \infty and \| (an)\| \infty = \mathrm{s}\mathrm{u}\mathrm{p}n\in \BbbN | an| < \infty , respectively. We denote by \sigma any sequence (zn) of points
in \BbbD . Recall that \sigma is said to be a Blaschke sequence if it satisfies the condition
\infty \sum
n=1
(1 - | zn| ) < \infty .
For a Blaschke sequence \sigma , the analytic function B = B\sigma defined in \BbbD by
B(z) =
\infty \prod
n=1
| zn|
zn
zn - z
1 - znz
is called the Blaschke product with zeros at \sigma and it is bounded by one. For a fixed n \in \BbbN , we denote
by Bn the Blaschke product B\sigma \setminus \{ zn\} . We put \rho (z, w) for the pseudohyperbolic distance between z,
w \in \BbbD , that is
\rho (z, w) =
\bigm| \bigm| \bigm| \bigm| z - w
1 - zw
\bigm| \bigm| \bigm| \bigm| .
Given a point \eta in the boundary of \BbbD and a number t \in [1,\infty ), the domain\bigl\{
z \in \BbbD / | 1 - \eta z| \leq t (1 - | z| )
\bigr\}
is called Stolz angle with vertex at \eta . A sequence \sigma is called uniformly separated if there exists \delta > 0
such that
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
| Bn(zn)| \geq \delta .
As usual, we will write c for positive constants when they appear. First, we recall two types of
distinguished sequences in \BbbD for H\infty :
c\bigcirc F. TUGORES, 2017
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2 279
280 F. TUGORES
Definition 1. \sigma = (zn) is called a zero sequence if there exists a function f \in H\infty , not
identically zero, such that f(zn) = 0 \forall n \in \BbbN .
Definition 2. \sigma = (zn) is called an interpolating sequence if given any sequence (an) \in l\infty ,
there exists a function f \in H\infty such that f(zn) = an \forall n \in \BbbN .
Clearly, interpolating sequences are zero sequences. It is well known that \sigma is a zero sequence
if and only if it is a Blaschke sequence [3]. On the other hand, it is proved in [2] that \sigma is an
interpolating sequence if and only if it is uniformly separated. Let \scrR be the space of real sequences
defined by
\scrR =
\bigl\{
(rn) / rn \in (0, 1) \forall n \in \BbbN and \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
rn = 0
\bigr\}
.
Next, we introduce another class of distinguished sequences in \BbbD for H\infty :
Definition 3. For a fixed sequence (rn) in \scrR , we say that \sigma = (zn) is a [rn]-reducing sequence
if given any sequence (an) \in l\infty , there exists a function f \in H\infty such that f(zn) = rn an \forall n \in \BbbN .
We will say that the function f in Definition 3 is a [rn]-reducing function. Note that if \sigma is a
[rn]-reducing sequence, then | f(zn) - an| = (1 - rn) | an| \forall n \in \BbbN , that is, each value f(zn) is on a
circle centred in an and radius strictly between 0 and | an| , approaching | an| . Since
\rho (f(zn), an) =
1 - rn
1 - rn | an| 2
| an| \forall n \in \BbbN ,
it follows that if (an) \subset \BbbD , then values f(zn) are also on pseudohyperbolic circles with the same
properties as before. If l\infty [rn] denotes the subspace of l\infty defined by
l\infty [rn] =
\bigl\{
(bn) \in l\infty / bn = rn an for a certain (an) \in l\infty
\bigr\}
,
then [rn]-reducing sequences can be viewed as interpolating sequences for the target space l\infty [rn].
Thus, it follows that every interpolating sequence is [rn]-reducing for any (rn) in \scrR .
We also state one variation of Definition 3:
Definition 4. For a fixed sequence (rn) in \scrR , we say that \sigma = (zn) is a [rn]-reducing sequence
in the l1-sense if given any sequence (an) \in l1, there exists a function f \in H\infty such that f(zn) =
= rn an \forall n \in \BbbN .
Clearly, every [rn]-reducing sequence is a [rn]-reducing sequence in the l1-sense. Taking (an) \equiv
(0) in Definitions 3 and 4, it turns out that all [rn]-reducing sequences and [rn]-reducing sequences in
the l1-sense are zero sequences. Then [rn]-reducing sequences are intermediate between interpolating
and zero sequences. The following section is devoted to examine both types of introduced sequences.
2. Statement and proof of results. First, we characterize [rn]-reducing sequences in the
l1-sense:
Theorem 1. A sequence \sigma = (zn) is [rn]-reducing in the l1-sense if and only if it is a Blaschke
sequence and the sequence (rn) in \scrR verifies
rn \leq c | Bn(zn)| \forall n \in \BbbN .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
REDUCING SEQUENCES 281
Proof. Necessity. Suppose that the restriction operator on \sigma maps H\infty onto the subspace of l1
defined by
l1[rn] =
\bigl\{
(bn) \in l1/ bn = rn an for a certain (an) \in l1
\bigr\}
,
with the norm \| (bn)\| l1[rn] = \| (an)\| 1. Fixed m \in \BbbN , we apply the open mapping theorem to the
sequence (bn) \in l1[rn] defined by means of (an) such that am = 1 and an = 0, if n \not = m. Then we
obtain that there exists fm \in H\infty such that
\| fm\| \infty \leq c \| (bn)\| l1[rn] = c \| (an)\| 1 = c.
Since fm = gmBm, for a certain gm not vanishing on \sigma and verifying \| gm\| \infty = \| fm\| \infty , it follows
that
rm = | fm(zm)| = | gm(zm)| | Bm(zm)| \leq \| gm\| \infty | Bm(zm)| \leq c | Bm(zm)| .
Sufficiency. For a sequence (an) in l1, we define the function f in \BbbD by
f(z) =
\infty \sum
n=1
rn an
Bn(z)
Bn(zn)
. (1)
Then f(zk) = rk ak \forall k \in \BbbN . On the other hand, f \in H\infty , because
| f(z)| \leq c
\infty \sum
n=1
| an| < \infty \forall z \in \BbbD .
Next, we put a condition on \sigma and another on (rn) in \scrR so that \sigma is [rn]-reducing:
Theorem 2. If \sigma = (zn) is a Blaschke sequence and (rn) in \scrR verifies
rn \leq c (1 - | zn| ) | Bn(zn)| \forall n \in \BbbN , (2)
then \sigma is [rn]-reducing and furthermore, any [rn]-reducing function f is Lipschitz on \sigma , that is,
| f(zn) - f(zm)| \leq c | zn - zm| \forall n, m \in \BbbN . (3)
Proof. For a sequence (an) in l\infty , the function f defined in (1) is in H\infty , because
| f(z)| \leq c
\infty \sum
n=1
rn
| Bn(zn)|
\leq c
\infty \sum
n=1
(1 - | zn| ) < \infty \forall z \in \BbbD .
On the other hand,
| f(zn) - f(zm)| \leq | f(zn)| + | f(zm)| = rn | an| + rm | am| \leq
\leq c \| (an)\| \infty [(1 - | zn| ) | Bn(zn)| + (1 - | zm) | Bm(zm)| ] \leq
\leq c [(1 - | zn| ) + (1 - | zm| )] \rho (zn, zm) \leq c | zn - zm| \forall n, m \in \BbbN .
If f \in H\infty , it is well known that
| f(z) - f(w)| \leq c \rho (z, w) \forall z, w \in \BbbD . (4)
Then, estimate (3) is an improvement of (4) on the sequence \sigma .
We are interested in having some control of the derivative of the [rn]-reducing function. Taking
a sequence (rn) in \scrR slightly more reductive than (3), we can obtain such control on the sequence
\sigma :
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
282 F. TUGORES
Theorem 3. If \sigma = (zn) is a Blaschke sequence and (rn) in \scrR verifies
rn \leq c (1 - | zn| ) | Bn(zn)| 2 \forall n \in \BbbN ,
then there is a [rn]-reducing function f such that
| (1 - | zn| ) f \prime (zn) - (1 - | zm| ) f \prime (zm)| \leq c | zn - zm| \forall n, m \in \BbbN . (5)
Proof. Given a sequence (an) in l\infty , we define the function f in \BbbD by
f(z) =
\infty \sum
n=1
rn an
Bn(z)
2
Bn(zn)2
.
Thus, f(zk) = rk ak \forall k \in \BbbN and it is immediate that f \in H\infty (it is proved as in Theorem 2). Since
f \prime (z) = 2
\infty \sum
n=1
rn an
Bn(z)B
\prime
n(z)
Bn(zn)2
,
then
f \prime (zk) = 2 rk ak
B\prime
k(zk)
Bk(zk)
\forall k \in \BbbN .
Taking into account that
| B\prime
k(zk)| \leq c
1
1 - | zk|
\forall k \in \BbbN ,
we have | f \prime (zk)| \leq c | Bk(zk)| \forall k \in \BbbN , and
| (1 - | zn| ) f \prime (zn) - (1 - | zm| ) f \prime (zm)| \leq (1 - | zn| ) | f \prime (zn)| + (1 - | zm| ) | f \prime (zm)| \leq
\leq c [(1 - | zn| ) | Bn(zn)| + (1 - | zm) | Bm(zm)| ] \leq c | zn - zm| \forall n, m \in \BbbN .
If f \in H\infty , it is proved in [1] that\bigm| \bigm| (1 - | z| ) f \prime (z) - (1 - | w| ) f \prime (w)
\bigm| \bigm| \leq c \rho (z, w) \forall z, w \in \BbbD . (6)
Then, estimate in (5) is an improvement of (6) on the sequence \sigma .
Adding one condition to the sequence \sigma , we can obtain a global control of the derivative of the
[rn]-reducing function:
Theorem 4. If \sigma = (zn) is a Blaschke sequence located in a Stolz angle and (rn) in \scrR verifies
the condition in (2), then there is a [rn]-reducing function f such that f \prime belongs to A1.
Proof. Given a sequence (an) in l\infty , we consider the function f defined in (1). Since
f \prime (z) =
\infty \sum
n=1
rn an
B\prime
n(z)
Bn(zn)
,
we have \int
\BbbD
| f \prime (z)| dm(z) \leq c
\int
\BbbD
\Biggl[ \infty \sum
n=1
(1 - | zn| ) | B\prime
n(z)|
\Biggr]
dm(z) =
= c
\infty \sum
n=1
(1 - | zn| )
\int
\BbbD
| B\prime
n(z)| dm(z).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
REDUCING SEQUENCES 283
It is proved in [4] that if \sigma is a Blaschke sequence located in a Stolz angle, then B\prime \in A1 and it
follows from there that
\| B\prime
n\| 1 < \| B\prime \| 1 \leq c \forall n \in \BbbN .
Thus, \int
\BbbD
| f \prime (z)| dm(z) \leq c
\infty \sum
n=1
(1 - | zn| ) < \infty .
Remark. We think that it would be interesting to pose reducing sequences in other spaces of
analytic functions, especially in the Bergman and Bloch spaces, to compare their results with those
obtained here.
References
1. Attele K. R. M. Interpolating sequences for the derivatives of Bloch functions // Glasgow Math. J. – 1992. – 34,
№ 1. – P. 35 – 41.
2. Carleson L. An interpolation problem for bounded analytic functions // Amer. J. Math. – 1958. – 80. – P. 921 – 930.
3. Duren P. L. Theory of Hp spaces. – New York; London: Acad. Press XII, 1970.
4. Girela D., Peláez J. A., Vukotić D. Integrability of the derivative of a Blaschke product // Proc. Edinburgh Math.
Soc. – 2007. – 50. – P. 673 – 687.
Received 17.04.15
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
|
| id | umjimathkievua-article-1694 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:47Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d5/bf783e348c46d180c7529a171ba3a5d5.pdf |
| spelling | umjimathkievua-article-16942019-12-05T09:23:56Z Reducing sequences Звiднi послiдовностi Tugores, F. Тугорес, Ф. We introduce and examine two new classes of distinguished sequences in the unit disk for the space of bounded analytic functions. One of these sequences is intermediate between interpolating and zero sequences. Введено та вивчено два нових класи видiлених послiдовностей в одиничному колi для простору обмежених аналiтичних функцiй. Одна з цих послiдовностей є промiжною мiж iнтерполяцiйною та нульовою послiдовностями. Institute of Mathematics, NAS of Ukraine 2017-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1694 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 2 (2017); 279-283 Український математичний журнал; Том 69 № 2 (2017); 279-283 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1694/676 Copyright (c) 2017 Tugores F. |
| spellingShingle | Tugores, F. Тугорес, Ф. Reducing sequences |
| title | Reducing sequences |
| title_alt | Звiднi послiдовностi |
| title_full | Reducing sequences |
| title_fullStr | Reducing sequences |
| title_full_unstemmed | Reducing sequences |
| title_short | Reducing sequences |
| title_sort | reducing sequences |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1694 |
| work_keys_str_mv | AT tugoresf reducingsequences AT tugoresf reducingsequences AT tugoresf zvidniposlidovnosti AT tugoresf zvidniposlidovnosti |