On the Berg-Chen-Ismail Theorem and the Nevanlinna-Pick Problem
In 2002, C. Berg, Y. Chen, and M. Ismail found a nice relation between the determinacy of the Hamburger moment problem and asymptotic behavior of the smallest eigenvalues of the corresponding Hankel matrices. We investigate whether an analog of this statement holds for the Nevanlinna-Pick interpolat...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2008
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irk-123456789-1065182016-09-30T03:03:03Z On the Berg-Chen-Ismail Theorem and the Nevanlinna-Pick Problem Golinskii, L. Peherstorfer, F. Yuditskii, P. In 2002, C. Berg, Y. Chen, and M. Ismail found a nice relation between the determinacy of the Hamburger moment problem and asymptotic behavior of the smallest eigenvalues of the corresponding Hankel matrices. We investigate whether an analog of this statement holds for the Nevanlinna-Pick interpolation problem. 2008 Article On the Berg-Chen-Ismail Theorem and the Nevanlinna-Pick Problem / L. Golinskii, F. Peherstorfer, P. Yuditskii // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 4. — С. 451-456. — Бібліогр.: 5 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106518 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In 2002, C. Berg, Y. Chen, and M. Ismail found a nice relation between the determinacy of the Hamburger moment problem and asymptotic behavior of the smallest eigenvalues of the corresponding Hankel matrices. We investigate whether an analog of this statement holds for the Nevanlinna-Pick interpolation problem. |
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Article |
| author |
Golinskii, L. Peherstorfer, F. Yuditskii, P. |
| spellingShingle |
Golinskii, L. Peherstorfer, F. Yuditskii, P. On the Berg-Chen-Ismail Theorem and the Nevanlinna-Pick Problem Журнал математической физики, анализа, геометрии |
| author_facet |
Golinskii, L. Peherstorfer, F. Yuditskii, P. |
| author_sort |
Golinskii, L. |
| title |
On the Berg-Chen-Ismail Theorem and the Nevanlinna-Pick Problem |
| title_short |
On the Berg-Chen-Ismail Theorem and the Nevanlinna-Pick Problem |
| title_full |
On the Berg-Chen-Ismail Theorem and the Nevanlinna-Pick Problem |
| title_fullStr |
On the Berg-Chen-Ismail Theorem and the Nevanlinna-Pick Problem |
| title_full_unstemmed |
On the Berg-Chen-Ismail Theorem and the Nevanlinna-Pick Problem |
| title_sort |
on the berg-chen-ismail theorem and the nevanlinna-pick problem |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/106518 |
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On the Berg-Chen-Ismail Theorem and the Nevanlinna-Pick Problem / L. Golinskii, F. Peherstorfer, P. Yuditskii // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 4. — С. 451-456. — Бібліогр.: 5 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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2025-07-07T18:35:54Z |
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1837014298040205312 |
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Journal of Mathematical Physics, Analysis, Geometry
2008, vol. 4, No. 4, pp. 451�456
On the Berg�Chen�Ismail Theorem
and the Nevanlinna�Pick Problem
L. Golinskii
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:leonid.golinskii@gmail.com
F. Peherstorfer and P. Yuditskii
Institute for Analysis, Johannes Kepler University Linz
A-4040 Linz, Austria
E-mail:franz.peherstorfer@jku.at
petro.yuditskiy@jku.at
Received May 19, 2008
In 2002, C. Berg, Y. Chen, and M. Ismail found a nice relation between
the determinacy of the Hamburger moment problem and asymptotic beha-
vior of the smallest eigenvalues of the corresponding Hankel matrices. We in-
vestigate whether an analog of this statement holds for the Nevanlinna�Pick
interpolation problem.
Key words: moment problem, Blaschke product, Carleson measure, Pick
matrix.
Mathematics Subject Classi�cation 2000: 30E05 (primary), 30D50
(secondary).
The Hamburger moment problem is a problem of �nding conditions on
a sequence fsjg, j = 0; 1; : : : ; so that there exists a positive Borel measure � and
sj =
Z
R
x
j
d�(x); j = 0; 1; : : : : (1)
With � one can associate the in�nite Hankel matrix and the sequence of its
principal submatrices
H = ksi+jk
1
i;j=0; Hn = ksi+jk
n
i;j=0; n = 0; 1; 2; : : : : (2)
Partially supported by the Austrian Founds FWF (Project P20413�N18) and Marie Curie
International Fellowship within the 6th European Community Framework Programme (Contract
MIF1-CT-2005-00696).
c
L. Golinskii, F. Peherstorfer, and P. Yuditskii, 2008
L. Golinskii, F. Peherstorfer, and P. Yuditskii
By the famous result of Hamburger (1) has a solution if and only if Hn � 0 for
all n.
Denote by f�n;jg
n
j=0 the eigenvalues of Hn (2) labelled in increasing order.
Because of the interlacing property we have
0 � �n+1;0 � �n;0 � �n+1;1 � �n;1 � : : : � �n+1;n � �n;n � �n+1;n+1; (3)
and so for each k �n;k are monotone decreasing, �k(H) = limn!1 �n;k exists, and
0 � �0(H) � �1(H) � : : : : (4)
Let us call the problem (1) regular if �0(H) > 0, and singular otherwise.
In 2002, C. Berg, Y. Chen, and M. Ismail [2] proved a beautiful result which states
that (1) is regular if and only if it has in�nitely many solutions (indeterminate).
The Hamburger moment problem is one of the representatives of the so-called
classical interpolation problems [1]. In this note we address to another one, speci�-
cally, the Nevanlinna�Pick interpolation problem in the Schur class S of functions
contractive and analytic in the unit disk D . This is a problem of �nding the
solutions of
f(zk) = wk; k = 0; 1; 2; : : : ; (5)
where zk are distinct points in D , wk complex numbers, and f 2 S. The well-
known criterion for (5), to have at least one solution, is given in terms of Pick
matrices by
Pn :=
1� wi �wj
1� zi�zj
n
i;j=0
� 0
for all n = 0; 1; : : : . For the eigenvalues f�n;jg
n
j=0 of the matrices Pn labelled
in increasing order the above relations (3), (4) hold, and, again, we distinguish
between the regular and singular Nevanlinna�Pick problem, i.e., in this paper we
say that the problem (5) is regular if �0(P ) > 0, and it is singular if �0(P ) = 0.
With respect to a number of various questions there is a strong similarity
between di�erent classical interpolation problems, that is, if one can prove this
or that statement with respect to one of the classical problems, a quite parallel
statement holds for another one. In this note we study the question: Is it true that
a Nevanlinna�Pick problem has in�nitely many solutions (indeterminate) if and
only if �0(P ) > 0? We give a negative answer to this question. More precisely,
we construct data fzk; wkg
1
k=0 of an indeterminate Nevanlinna�Pick problem such
that �0(P ) = 0.
First of all, we note that the Blaschke condition on the interpolation nodes
Z = fzkg
1X
k=0
(1� jzkj) <1 (6)
452 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On the Berg�Chen�Ismail Theorem and the Nevanlinna�Pick Problem
guarantees that the interpolation problem
f(zk) = 0; k = 0; 1; 2; : : : ; (7)
has in�nitely many solutions. Indeed, every function of the form
f = g(z)B(z);
where g(z) 2 S, and B(z) is the Blaschke product
B(z) =
Y
k
zk � z
1� z�zk
jzkj
zk
;
solves (7). Note that (6) is necessary and su�cient for the problem (7) to be
indeterminate.
Thus, our goal is to construct a Blaschke set Z such that �0(P ) = 0 for the
sequence of Nevanlinna�Pick matrices of the speci�c form Pn = Kn, where
Kn :=
1
1� zi�zj
n
i;j=0
:
In fact, our main statement here characterizes completely the regularity of such
Nevanlinna�Pick problems in the above sense.
Recall (see, e.g., [4]), that a (�nite) Borel measure � on D is called a Carleson
measure, if Z
D
jf j2 d� � C
Z
T
jf j2 dm (8)
for all f 2 H
2. Here dm is the normalized Lebesgue measure on the unit circle T.
Due to Carleson's theorem [3] such measures are characterized completely by the
following property: there exists C > 0 such that
�(Q�(�)) � C�
for all �� < � � �, 0 < � < 1,
Q�(�) := fz 2 D : j arg z � �j � ��; 1� � � jzj < 1g:
Theorem 1. Let Z satisfy (6), B(z) be the corresponding Blaschke product.
The Nevanlinna�Pick problem (7) is regular if and only if the measure �, de�ned by
�(fzkg) = jB0(zk)j
�2
; (9)
is a Carleson measure in D .
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 453
L. Golinskii, F. Peherstorfer, and P. Yuditskii
P r o o f. Let � (9) be a Carleson measure. For arbitrary c0; : : : ; cn 2 C put
h(z) = B(z)
nX
k=0
ck
z � zk
2 H
2
with
khk2 =
nX
i;j=0
Kijcicj ; K = kKijk
1
i;j=0 =
1
1� zi�zj
1
i;j=0
:
We have h(zj) = cjB
0(zj) for j = 0; 1; : : : ; n and h(zj) = 0 for j � n+ 1, so
Z
D
jhj2 d� =
nX
j=0
jcj j
2
:
Hence by (8) Z
T
jhj2 dm = khk2 �
1
C
nX
j=0
jcj j
2
;
and so �0(K) � C
�1
> 0, as claimed.
Conversely, assume that the Nevanlinna�Pick problem in question is regular.
We use the standard orthogonal decomposition L
2(T) = H
2
L
H
2
�
. Put
KB = (BH2)? = H
2 \BH2
�
; �k(z) =
B(z)
z � zk
; k = 0; 1; : : : :
It is easy to see that f�kg is complete in KB . Indeed, (z � zk)
�1 2 H
2
�, so
�k 2 KB . Let g 2 KB , g?�k for all k = 0; 1; : : : . Then g = B�g1(�), g1 2 H
2,
and
0 = (g; �k) =
Z
T
B(�)
� � zk
�B(�)�g1(�) dm = g1(zk);
that is, g1 2 BH
2, so g 2 H
2
�
and g � 0, as claimed. Hence the system of
functions
f(z) = B(z)
nX
k=0
ck
z � zk
+B(z)g(z) = h(z) +B(z)g(z); n = 0; 1; : : : ;
c0; : : : ; cn 2 C , and g 2 H
2, is dense in H
2. We prove (8) for such functions.
As above,
f(zj) = cjB
0(zj); j = 0; 1; : : : ; n; f(zj) = 0; j � n+ 1;
454 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On the Berg�Chen�Ismail Theorem and the Nevanlinna�Pick Problem
and Z
D
jf j2 d� =
nX
j=0
jcj j
2
; kfk2 = khk2 + kgk2 � khk2 =
nX
i;j=0
Kijcicj ;
and so Z
T
jf j2 dm �
nX
i;j=0
Kijcicj � �0(K)
nX
j=0
jcj j
2
= �0(K)
Z
D
jf j2 d�:
The proof is complete.
If the Nevanlinna�Pick problem (7) is singular (as in the example below), then
so is the general problem (5). Indeed, for D = diag(w0; w1; : : : ; wn)
Pn = Kn �DKnD
� � Kn;
and so �0(P ) � �0(K).
E x a m p l e. Put zk = 1�k�p, p > 1. Evidently Z is a Blaschke set. On the
other hand
(1� jznj
2)jB0(zn)j =
n�1Y
k=1
zn � zk
1� znzk
1Y
k=n+1
zk � zn
1� znzk
�
1Y
k=n+1
k
p � n
p
np + kp � 1
�
1Y
k=n+1
�
1�
�
n
k
�p�
� exp
�
1X
k=n+1
�
n
k
�p!
� exp
�
�
n+ 1
2p(p� 1)
�
;
so
jB0(zn)j
�2 �
1
n2p
exp
�
n+ 1
2p�1(p� 1)
�
:
Thus the measure � (9) is in�nite and, moreover, it is not of Carleson type.
There is a simple way of manufacturing regular Nevanlinna�Pick problems
(7). Recall that Z = fzkg is the Carleson (uniformly separated) sequence if
Æ(Z) := inf
n
������
Y
k 6=n
jzkj
zk
zn � zk
1� �znzk
������ > 0: (10)
Assume that Z satis�es (10). By the theorem of H.S. Shapiro and A.L. Shields
[5] the system of functions
xk(z) =
(1� jzkj
2)1=2
1� �zkz
k = 0; 1; : : : ;
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 455
L. Golinskii, F. Peherstorfer, and P. Yuditskii
forms a Riesz basis in KB . So, for all c0; : : : ; cn 2 C there is c > 0 such that
nX
i;j=0
Kij(1� jzij
2)1=2 (1� jzj j
2)1=2 cicj � c
nX
j=0
jcj j
2
;
or
nX
i;j=0
Kij didj � c
nX
j=0
jdj j
2
(1� jzj j2)
� c
nX
j=0
jdj j
2
;
as claimed.
Acknowledgement. The authors thank Alexander Kheifetz for helpful dis-
cussions. The paper is written mainly during the �rst author's visit to Johannes
Kepler University, Linz. He wishes to thank JKU for the hospitality and the
Marie Curie Foundation that made this visit possible.
References
[1] N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in
Analysis. Oliver and Boyed, Edinburgh, 1965.
[2] C. Berg, Y. Chen, and M. Ismail, Small Eigenvalues of Large Hankel Matrices:
the Indeterminate Case. � Math. Scand. 91 (2002), No. 1, 67�81.
[3] L. Carleson, An Interpolation Problem for Bounded Analytic Functions. � Amer.
J. Math. 80 (1958), No. 4, 921�930.
[4] J.B. Garnett, Bounded Analytic Functions. (Revised �rst edition.) Graduate Texts
in Mathematics. 236. Springer, New York, 2007. xiv+459 pp.
[5] H.S. Shapiro and A.L. Shields, On Some Interpolation Problems for Analytic
Functions. � Amer. J. Math. 83 (1961), No. 3, 513�532.
456 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
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