Nevanlinna matrices for the strong Stieltjes moment problem
Olav Njastad∗
Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7034 Trondheim, Norway
Received 30 October 1997; received in revised form 15 March 1998
Abstract
Let {cn}
∞
n=−∞ be a doubly innite sequence of real numbers. A solution of the strong Hamburger moment problem is
a positive measure on (−∞;∞) such thatcn=R
∞
−∞u
nd(u) forn= 0;±1;±2; : : : :A solution of the strong Stieltjes moment problem is a positive measureon [0;∞) such thatcn=R
∞
0 u
nd(u) forn= 0;±1;±2; : : : :A moment problem is indeterminate if there exists more than one solution. With an indeterminate strong Hamburger moment problem there is associated a Nevanlinna matrix of functions; ; ; holomorphic inC− {0}. These functions have growth properties partly similar to properties of analogous entire functions associated with an indeterminate classical Hamburger moment problem. In this paper we obtain a stronger growth result in the case where the strong Stieltjes moment problem is solvable. c1998 Elsevier Science B.V. All rights reserved.
AMS classication:30D15; 30E05; 42C05; 44A60
Keywords:Strong moment problems; Nevanlinna parametrization; Nevanlinna matrices
1. Introduction
The Hamburger Moment Problem(HMP) may be described as follows: Let{cn}∞n=0 be a sequence
of real numbers. Find conditions for the existence and uniqueness of measures satisfying
cn=
Z ∞
−∞
und(u) for n= 0;1;2; : : : ; (1.1)
and study structures connected with the set of solutions. A solution of the Stieltjes Moment Problem (SMP) for the sequence {cn} is a solution of the HMP whose support is contained in the interval
[0;∞).
A moment problem is called determinate if there is exactly one solution, indeterminateif there is more than one solution. In the indeterminate case of the HMP there exist entire functions A; B; C; D
∗
E-mail: njastad@imf.unit.no.
such that the formula
ˆ
(z) =−A(z)’(z)−C(z)
B(z)’(z)−D(z) (1.2)
determines a one-to-one correspondence between all solutions of the HMP and all Pick functions
’. Here a Pick function is a function’ holomorphic in the open upper half-planeU and mapping U
into the closure ofU on the Riemann sphere, the constant∞ included, while ˆ denotes the Stieltjes
transform of :
ˆ
(z) = Z ∞
−∞ d(u)
u−z : (1.3)
The functions A; B; C; Dare entire transcendental functions of at most minimal type of order 1. That an entire function f is of at most minimal type of order 1 means that for every positive number ”
there exists a constant M such that
|f(z)|6Me”|z| for all z∈C: (1.4)
(C denotes the nite complex plane.)
For detailed treatments of the classical moment problems, see [1, 3, 5–7, 9, 13–15, 22–26]. The Strong Hamburger Moment Problem (SHMP) is dened like the classical problem except that the given sequence is a doubly innite sequence {cn}∞n=−∞ and the equality (1.1) is required to hold for n= 0;±1;±2; : : : :In the indeterminate case there exist in this situation functions ; ; ;
which are holomorphic in C− {0} such that the formula
ˆ
(z) =− (z)’(z)−(z)
(z)’(z)−(z) (1.5)
determines a one-to-one correspondence between all solutions of the SHMP and all Pick func-tions ’. See [19].
We showed in [20] that these functions ; ; ; satisfy an inequality of the form
|f(z)|6Me”
|z|+|1z|
: (1.6)
However, we were only able to show the validity of such an inequality in every region given by
6|argz|6−; ¿0, with M depending on (and ”).
In this paper we show that if the SHMP admits a solution supported in [0;∞), i.e. if the Strong Stieltjes Moment Problem (SSMP) is solvable, then ; ; and satisfy an inequality of the form (1.6) in every angular region given by |argz|6−; ¿0.
For earlier work on strong moment problems, see [2, 4, 8, 10–12, 16–21].
2. Orthogonal Laurent polynomials
A Laurent polynomial is an element in the linear spacespanned by the monomialszn; n= 0;±1;
±2: : : : Let S be the linear functional dened on this space by
S[zn] =c
A necessary and sucient condition for the SHMP to be solvable is that S is positive on R, while
a necessary and sucient condition for the SSMP to be solvable is that S is positive on R+. (S is said to be positive in an interval I if S(L)¿0 for all L∈ where L(z)6≡0; L(z)¿0 for z∈I.) We shall in the following always assume that S is positive on R;and thus that the SHMP is solvable;
with all solutions having innite support. (See e.g. [10].)
A (non-degenerate) inner product h;i is dened on the space of real Laurent polynomials by
hf; gi=S[f(x)g(x)]: (2.2) By orthonormalization of the basis{1; z−1; z; z−2; z2; : : : ; z−m; zm; : : :} we obtain an orthonormal system
of Laurent polynomials {’n}∞n=0. These Laurent polynomials may be written in the form
’2m(z) =
some technical complications arise in the treatment of the SHMP, but most of the main results are valid in all cases. WhenS is positive onR+, and thus the SSMP is solvable, the sequence is always
regular. For convenience we shall in the following assume that the sequence {’n}∞n=0 is regular also
when positivity of S on R+ is not assumed.
The associated orthogonal Laurent polynomials n are dened by
n(z) =S
(the functional operating on its argument as a function of u).
For further reference we state as a proposition the following result on the zeros of ’n and n.
(See [10, 12, 17, 19].)
Proposition 2.1. The zeros of ’n and of n are real and simple; and between any two positive (or
These functions are real Laurent polynomials. It follows from Christoel–Darboux formulas for the orthogonal Laurent polynomials that 2m(z); 2m(z) are quasi-orthogonal Laurent polynomials and 2m(z); 2m(z) are associated quasi-orthogonal Laurent polynomials, while z−12m+1(z); z−12m+1(z)
are quasi-orthogonal Laurent polynomials and z−1
2m+1(z); z−12m+1(z) are associated
quasi-orthogonal Laurent polynomials. See [17, 19, 21]. (Quasi-quasi-orthogonal and associated quasi-quasi-orthogonal Laurent polynomials are functions of the form ’n(z)−z(−1)
tradicts the determinant formula for orthogonal and associated orthogonal Laurent polynomials, see e.g. [19, 21]. Thus the two abovementioned Laurent polynomials can have no common zero. Let
() and () be two consecutive positive zeros of ’n(z)−z(−1)
zeros with respect to and the impossibility of common zeros established above, it follows that for any given there is a zero () of n(z)−z(−1)
For more information on orthogonal and quasi-orthogonal Laurent polynomials, see [10, 16–21].
3. Mobius transformations
We dene the quasi-approximants Tn(z; t) associated with the moment sequence by
Tn(z; t) =
maps the open upper half-plane U onto an open disk n(z) and the boundary ˆR onto the boundary
(Recall the denition (1.3) of the Stieltjes transform ˆ.) Thus the SHMP is determinate in the limit point case, indeterminate in the limit circle case.
In the limit point case the series P∞
n=0|’n(z)|2 and
P∞
n=0| n(z)|2 diverge for all z∈C−{R}, while
in the limit circle case these series converge locally uniformly in C− {0}. The radius (z) of the disk ∞(z) is given by
(z) = "
|z−z|
∞
X
n=0
|’n(z)|2
#−1
: (3.4)
Furthermore, in the limit circle case the sequences{n}; {n}; {n}; {n}converge locally uniformly
in C− {0} to functions ; ; ; . These functions are then holomorphic in C− {0}, and give rise to
the correspondence (Nevanlinna parametrization) described in (1.5). They satisfy the equality
(z)(z)−(z)(z) = 1: (3.5)
The mapping
t→ − (z)t−(z)
(z)t−(z) (3.6)
maps the closed upper half-plane U∪ Rˆ onto the disk ∞(z), the open upper half-plane U onto the
interior ∞(z) and the extended real line ˆR onto the cicumference @∞(z). For more information on the topics treated in this section, see [11, 16–21].
4. Nevanlinna matrices
In analogy with the classical situation (see e.g. [1]) we call a matrix of the form
a(z) c(z)
b(z) d(z)
(4.1)
a Nevanlinna matrix of functions holomorphic in C− {0} if a; b; c; d are functions holomorphic in C− {0} with essential singularities at the origin, which satisfy
a(z)d(z)−b(z)c(z) = 1 (4.2)
in C− {0} and where for each t∈ Rˆ the function −[a(z)t−c(z)]=[b(z)t−d(z)] is holomorphic for z∈C−R and maps U into U, −U into −U.
The special functions ; ; ; occuring in connection with an indeterminate SHMP as described in Section 3, constitute such a Nevanlinna matrix. In [20] we studied growth properties of the functions
; ; ; and also 1=.
We dene for ∈(0;=2) the angular region by
={z∈C− {0}: 6|argz|6−}: (4.3)
Theorem 4.1. For arbitrary xed numbers and ”; 0¡¡=2; ”¿0; there exists a constant
The main purpose of this paper is to strenghten this result in the situation where the corresponding SSMP is solvable. We dene for ∈(0;) the angular region by
={z∈C− {0}:|argz|6−}: (4.5)
We shall prove the following result.
Letting ntend to innity and combining the result with (4.4) for |argz|=we get for all z∈ −:
|f(z)|6√1
cosM(; ”)e
”(R+(1=R))= 1
√cos M(; ”)e”(|z|+(1=|z|)): (4.12)
We conclude from (4.12) and Theorem 4.1 that
|f(z)|6√1
cosM(; ”)e
”(|z|+(1=|z|) for z
∈ ; (4.13)
where f is one of the functions ; ; ; .
We recall from Proposition 2.1 that all the zeros of ’n are positive. It follows that |’n(Rei)|2 is
increasing as a function of ||, and hence from the denition (3.4) that 1=(Rei) is increasing as a
function of ||. Combining this with (4.4) for |argz|= we get for all z∈ −:
1
(z)6 1
(Rei)6M(; ”)e
”(R+(1=R))=M(; ”)e”(|z|+(1=|z|)): (4.14)
We conclude from (4.14) and Theorem 4.1 that
1
(z)6M(; ”)e
”(|z|+1=|z|) for z
∈ : (4.15)
The desired result now follows from (4.13) and (4.15).
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