Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:
Journal of Computational and Applied Mathematics 99 (1998) 311–318
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 in nite sequence of Rreal numbers. A solution of the strong Hamburger moment problem is
∞
a positive measure on (−∞; ∞) such that cn = −∞ u n d(u) for n = 0; ± 1; ± 2; : : : : A solution of the strong Stieltjes
R∞
moment problem is a positive measure on [0; ∞) such that cn = 0 u n d(u) for n = 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 in C − {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
c 1998 Elsevier Science B.V. All rights reserved.
solvable.
AMS classi cation: 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
∞
u n d(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, indeterminate if there is
more than one solution. In the indeterminate case of the HMP there exist entire functions A; B; C; D
∗
E-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 6 5 - 4
312
O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
such that the formula
A(z)’(z) − C(z)
(z)
ˆ =−
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-plane U and mapping U
into the closure of U on the Riemann sphere, the constant ∞ included, while ˆ denotes the Stieltjes
transform of :
(z)
ˆ =
Z
∞
−∞
d(u)
:
u−z
(1.3)
The functions A; B; C; D are 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)|6M e”|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 de ned like the classical problem except that
the given sequence is a doubly in nite 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
(z)
ˆ =−
(z)’(z) −
(z)
(z)’(z) − (z)
(1.5)
determines a one-to-one correspondence between all solutions of the SHMP and all Pick functions ’. See [19].
We showed in [20] that these functions ; ;
; satisfy an inequality of the form
|f(z)|6M e
” |z|+
1
|z|
:
(1.6)
However, we were only able to show the validity of such an inequality in every region given by
6|arg z|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 |arg z|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 space spanned by the monomials z n ; n = 0; ± 1;
± 2 : : : : Let S be the linear functional de ned on this space by
S[z n ] = cn
for n = 0; ± 1; ± 2 : : : :
(2.1)
313
O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
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 in nite support. (See e.g. [10].)
A (non-degenerate) inner product h ; i is de ned 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 ; z 2 ; : : : ; z −m ; z m ; : : :} we obtain an orthonormal system
of Laurent polynomials {’n }∞
n=0 . These Laurent polynomials may be written in the form
q2m; −m
+ · · · + q2m; m z m ; q2m; m ¿0;
(2.3)
’2m (z) =
zm
q2m+1; −(m+1)
’2m+1 (z) =
+ · · · + q2m+1; m z m ; q2m+1; −(m+1) ¿0:
(2.4)
z m+1
The sequence {’n }∞
n=0 is called regular if q2m;−m 6= 0; q2m+1; m 6= 0 for all m. For nonregular sequences
some technical complications arise in the treatment of the SHMP, but most of the main results are
valid in all cases. When S is positive on R+ , 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 de ned by
n (z) = S
’n (u) − ’n (z)
u−z
(2.5)
(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
(See [10, 12, 17, 19].)
Proposition 2.1. The zeros of ’n and of
negative) zeros of ’n there is a zero of
n are positive.
n.
are real and simple; and between any two positive (or
.
n When S is positive on R+ ; all the zeros of ’n and of
n
Let x0 be an arbitrary xed point in R − {0}. We de ne functions n ; n ;
n ; n by
n (z) = (z − x0 )
n−1
X
k (x0 ) k (z);
n (z) = − 1 + (z − x0 )
n (z) = 1 + (z − x0 )
n (z) = (z − x0 )
(2.6)
k=0
n−1
X
k=0
n−1
X
k (x0 )’k (z);
(2.7)
k=0
n−1
X
’k (x0 ) k (z);
(2.8)
k=0
’k (x0 )’k (z):
(2.9)
314
O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
These functions are real Laurent polynomials. It follows from Christo el–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 −1 2m+1 (z); z −1 2m+1 (z)
are quasi-orthogonal Laurent polynomials and z −1 2m+1 (z); z −1
2m+1 (z) are associated quasiorthogonal Laurent polynomials. See [17, 19, 21]. (Quasi-orthogonal and associated quasi-orthogonal
n
n
Laurent polynomials are functions of the form ’n (z) − z (−1) ’n−1 (z) or n (z) − z (−1) n−1 (z);
∈ R̂ = R ∪ {∞}:)
The following result on the zeros of n ; n ;
n ; n will be used later.
Proposition 2.2. The zeros of the functions n ; n ;
n ; n are real and simple. If S is positive on R+ ;
then each of the functions n ; n ;
n ; n has at most one negative zero.
n
Proof. It is proved in [17] that the quasi-orthogonal polynomial ’n (z) − z (−1) ’n−1 (z) has only
real and simple zeros, at most one of them negative. For a possible common zero of ’n (z) −
n
n
z (−1) ’n−1 (z) and n (z) − z (−1) n−1 (z) we would have n ()’n−1 () − n−1 ()’n () = 0. This contradicts 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
n
( ) and ( ) be two consecutive positive zeros of ’n (z) − z (−1) ’n−1 (z). By Proposition 2.1 there
n
is a zero ( ) of n (z) − z (−1) n−1 (z) between ( ) and ( ) when = 0. By the continuity of the
zeros with respect to and the impossibility of common zeros established above, it follows that for
n
any given there is a zero ( ) of n (z) − z (−1) n−1 (z) between the positive zeros ( ) and ( )
(−1) n
(−1) n
of ’n (z) − z
’n−1 (z). Thus n (z) − z
n−1 (z) can have at most one negative zero.
For more information on orthogonal and quasi-orthogonal Laurent polynomials, see [10, 16–21].
3. Mobius transformations
We de ne the quasi-approximants Tn (z; t) associated with the moment sequence by
Tn (z; t) =
n (z)t −
n (z)
:
n (z)t − n (z)
(3.1)
For each z ∈ C − R the mapping
t → w = − Tn (z; t)
(3.2)
maps the open upper half-plane U onto an open disk n (z) and the boundary R̂ onto the boundary
circle @n (z). We denote the closed disk n (z) ∪ @n (z) by n (z).
T
The sequence of disks {n (z)} is nested, and thus the intersection ∞ (z) = ∞
n=1 n (z) is either a
single point or a proper closed disk. Moreover, ∞ (z) is either a single point for every z ∈ C − R
(limit point case) or a proper disk for every z ∈ C − R (limit circle case).
For each z ∈ C − R we have
∞ (z) = {w = (z):
ˆ
is a solution of the SHMP }:
(3.3)
O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
315
(Recall the de nition (1.3) of the Stieltjes transform .)
ˆ Thus the SHMP is determinate in the limit
point case, indeterminate in the limit
circle
case.
P
P∞
2
2
In the limit point case the series ∞
n=0 |’n (z)| and
n=0 | n (z)| 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 ∪ R̂ onto the disk ∞ (z), the open upper half-plane U onto the
interior ∞ (z) and the extended real line R̂ 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)
b(z)
c(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 ∈ R̂ 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 de ne for ∈ (0; =2) the angular region
by
= {z ∈ C − {0}: 6|arg z|6 − }:
A central result of [20] is the following theorem, valid for an arbitrary indeterminate SHMP.
(4.3)
316
O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
Theorem 4.1. For arbitrary xed numbers and ”; 0¡¡=2; ”¿0; there exists a constant
M (; ”) such that
1
|f(z)|6M (; ”)exp ” |z| +
|z|
for all z∈
;
(4.4)
where f is any of the functions ; ;
; ; 1=.
The main purpose of this paper is to strenghten this result in the situation where the corresponding
SSMP is solvable. We de ne for ∈ (0; ) the angular region by
= {z ∈ C − {0}: |arg z|6 − }:
(4.5)
We shall prove the following result.
Theorem 4.2. Assume that the SSMP associated with the strong moment sequence is solvable.
Then for arbitrary xed numbers and ”; 0¡¡; ”¿0; there exists a constant H (; ”) such that
|f(z)|6H (; ”)e
1
” |z|+| |
z
for all z ∈
;
(4.6)
where f is any of the functions ; ;
; ; 1=.
Proof. Set z = Rei and consider the function hR () = |z − |2 where is some xed number in R.
We nd that hR () = R2 − 2R cos + 2 . Thus hR () is increasing with increasing || when ∈ R+ ,
decreasing with increasing || when ∈ R− . Choose some ∈ (0; =2), and consider the function hR
for some ∈ R− , e.g. = re−i . We have
hR (0) = (R + r)2 ;
Thus for z ∈
hR () = r 2 + R2 + 2rR cos :
(4.7)
−
,
(r + R)2
hR () hR (0)
6
=
hR () hR () (r + R)2 − 2rR(1 − cos )
(4.8)
hR ()
1
:
6
hR () cos
(4.9)
and since 2rR6(r + R)2 =2 we get
This means that
|z − |6 √
1
|Rei − | for z ∈
cos
−
(4.10)
when ∈ R− .
Let fn denote one of the functions n ; n ;
n ; n . We know from Proposition 2.2 that fn has at
most one negative zero. By using (4.10) and the fact that |z − | is increasing as a function of ||
for ∈ R+ , we nd that
1
fn (Rei )
|fn (z)|6 √
cos
for z ∈
−
:
(4.11)
O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
Letting n tend to in nity and combining the result with (4.4) for |arg z| = we get for all z ∈
|f(z)|6 √
1
1
M (; ”)e”(R+(1=R)) = √
M (; ”)e”(|z|+(1=|z|)) :
cos
cos
317
−
:
(4.12)
We conclude from (4.12) and Theorem 4.1 that
|f(z)|6 √
1
M (; ”)e”(|z|+(1=|z|)
cos
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 de nition (3.4) that 1=(Rei ) is increasing as a
function of ||. Combining this with (4.4) for |arg z| = we get for all z ∈ −
:
1
1
6
6M (; ”)e”(R+(1=R)) = M (; ”)e”(|z|+(1=|z|)) :
(z) (Rei )
(4.14)
We conclude from (4.14) and Theorem 4.1 that
1
6M (; ”)e”(|z|+1=|z|)
(z)
for z ∈
:
(4.15)
The desired result now follows from (4.13) and (4.15).
References
[1] N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York, 1965.
[2] E. Alden, A survey of weak and strong moment problems with generalizations, Department of Mathematics,
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[3] C. Berg, H.L. Pedersen, On the order and type of the entire functions associated with an indeterminate Hamburger
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with discrete distributions, in: W.B. Jones, A. Sri Ranga (Eds.), Orthogonal Functions, Moment Theory and Continued
Fractions: Theory and Applications, Marcel Dekker, New York, 1998, 31–55.
[5] T.S. Chihara, On indeterminate Hamburger moment problems, Paci c J. Math. 27 (1968) 475– 484.
[6] T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
[7] T.S. Chihara, Indeterminate symmetric moment problems, J. Math. Anal. Appl. 85 (1982) 331–346.
[8] L. Cochran, S. Clement Cooper, Orthogonal Laurent polynomials on the real line, in: S. Clement Cooper, W.J. Thron
(Eds.), Continued Fractions and Orthogonal Functions, Marcel Dekker, New York, 1994, pp. 47–100.
[9] H. Hamburger, Uber
eine Erweiterung des Stieltjesschen Momentproblems, Parts I, II, III, Math. Ann. 81 (1920)
235–319; 82 (1921) 120–164; 168–187.
[10] W.B. Jones, O. Njastad, W.J. Thron, Orthogonal Laurent polynomials and the strong Hamburger moment problem,
J. Math. Anal. Appl. 98 (1984) 528–554.
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Math. Soc. 47 (1983) 363–384.
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503–528.
[13] H.J. Landau, The classical moment problem: Hilbertian proofs, J. Funct. Anal. 38 (1980) 255–272.
[14] H.J. Landau (Ed.), Moments in Mathematics, Proc. Symp. Appl. Math. 37, Amer. Math. Soc., Providence, RI, 1987.
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vol. 15, New York, 1932.
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 in nite sequence of Rreal numbers. A solution of the strong Hamburger moment problem is
∞
a positive measure on (−∞; ∞) such that cn = −∞ u n d(u) for n = 0; ± 1; ± 2; : : : : A solution of the strong Stieltjes
R∞
moment problem is a positive measure on [0; ∞) such that cn = 0 u n d(u) for n = 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 in C − {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
c 1998 Elsevier Science B.V. All rights reserved.
solvable.
AMS classi cation: 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
∞
u n d(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, indeterminate if there is
more than one solution. In the indeterminate case of the HMP there exist entire functions A; B; C; D
∗
E-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 6 5 - 4
312
O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
such that the formula
A(z)’(z) − C(z)
(z)
ˆ =−
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-plane U and mapping U
into the closure of U on the Riemann sphere, the constant ∞ included, while ˆ denotes the Stieltjes
transform of :
(z)
ˆ =
Z
∞
−∞
d(u)
:
u−z
(1.3)
The functions A; B; C; D are 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)|6M e”|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 de ned like the classical problem except that
the given sequence is a doubly in nite 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
(z)
ˆ =−
(z)’(z) −
(z)
(z)’(z) − (z)
(1.5)
determines a one-to-one correspondence between all solutions of the SHMP and all Pick functions ’. See [19].
We showed in [20] that these functions ; ;
; satisfy an inequality of the form
|f(z)|6M e
” |z|+
1
|z|
:
(1.6)
However, we were only able to show the validity of such an inequality in every region given by
6|arg z|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 |arg z|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 space spanned by the monomials z n ; n = 0; ± 1;
± 2 : : : : Let S be the linear functional de ned on this space by
S[z n ] = cn
for n = 0; ± 1; ± 2 : : : :
(2.1)
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O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
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 in nite support. (See e.g. [10].)
A (non-degenerate) inner product h ; i is de ned 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 ; z 2 ; : : : ; z −m ; z m ; : : :} we obtain an orthonormal system
of Laurent polynomials {’n }∞
n=0 . These Laurent polynomials may be written in the form
q2m; −m
+ · · · + q2m; m z m ; q2m; m ¿0;
(2.3)
’2m (z) =
zm
q2m+1; −(m+1)
’2m+1 (z) =
+ · · · + q2m+1; m z m ; q2m+1; −(m+1) ¿0:
(2.4)
z m+1
The sequence {’n }∞
n=0 is called regular if q2m;−m 6= 0; q2m+1; m 6= 0 for all m. For nonregular sequences
some technical complications arise in the treatment of the SHMP, but most of the main results are
valid in all cases. When S is positive on R+ , 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 de ned by
n (z) = S
’n (u) − ’n (z)
u−z
(2.5)
(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
(See [10, 12, 17, 19].)
Proposition 2.1. The zeros of ’n and of
negative) zeros of ’n there is a zero of
n are positive.
n.
are real and simple; and between any two positive (or
.
n When S is positive on R+ ; all the zeros of ’n and of
n
Let x0 be an arbitrary xed point in R − {0}. We de ne functions n ; n ;
n ; n by
n (z) = (z − x0 )
n−1
X
k (x0 ) k (z);
n (z) = − 1 + (z − x0 )
n (z) = 1 + (z − x0 )
n (z) = (z − x0 )
(2.6)
k=0
n−1
X
k=0
n−1
X
k (x0 )’k (z);
(2.7)
k=0
n−1
X
’k (x0 ) k (z);
(2.8)
k=0
’k (x0 )’k (z):
(2.9)
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O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
These functions are real Laurent polynomials. It follows from Christo el–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 −1 2m+1 (z); z −1 2m+1 (z)
are quasi-orthogonal Laurent polynomials and z −1 2m+1 (z); z −1
2m+1 (z) are associated quasiorthogonal Laurent polynomials. See [17, 19, 21]. (Quasi-orthogonal and associated quasi-orthogonal
n
n
Laurent polynomials are functions of the form ’n (z) − z (−1) ’n−1 (z) or n (z) − z (−1) n−1 (z);
∈ R̂ = R ∪ {∞}:)
The following result on the zeros of n ; n ;
n ; n will be used later.
Proposition 2.2. The zeros of the functions n ; n ;
n ; n are real and simple. If S is positive on R+ ;
then each of the functions n ; n ;
n ; n has at most one negative zero.
n
Proof. It is proved in [17] that the quasi-orthogonal polynomial ’n (z) − z (−1) ’n−1 (z) has only
real and simple zeros, at most one of them negative. For a possible common zero of ’n (z) −
n
n
z (−1) ’n−1 (z) and n (z) − z (−1) n−1 (z) we would have n ()’n−1 () − n−1 ()’n () = 0. This contradicts 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
n
( ) and ( ) be two consecutive positive zeros of ’n (z) − z (−1) ’n−1 (z). By Proposition 2.1 there
n
is a zero ( ) of n (z) − z (−1) n−1 (z) between ( ) and ( ) when = 0. By the continuity of the
zeros with respect to and the impossibility of common zeros established above, it follows that for
n
any given there is a zero ( ) of n (z) − z (−1) n−1 (z) between the positive zeros ( ) and ( )
(−1) n
(−1) n
of ’n (z) − z
’n−1 (z). Thus n (z) − z
n−1 (z) can have at most one negative zero.
For more information on orthogonal and quasi-orthogonal Laurent polynomials, see [10, 16–21].
3. Mobius transformations
We de ne the quasi-approximants Tn (z; t) associated with the moment sequence by
Tn (z; t) =
n (z)t −
n (z)
:
n (z)t − n (z)
(3.1)
For each z ∈ C − R the mapping
t → w = − Tn (z; t)
(3.2)
maps the open upper half-plane U onto an open disk n (z) and the boundary R̂ onto the boundary
circle @n (z). We denote the closed disk n (z) ∪ @n (z) by n (z).
T
The sequence of disks {n (z)} is nested, and thus the intersection ∞ (z) = ∞
n=1 n (z) is either a
single point or a proper closed disk. Moreover, ∞ (z) is either a single point for every z ∈ C − R
(limit point case) or a proper disk for every z ∈ C − R (limit circle case).
For each z ∈ C − R we have
∞ (z) = {w = (z):
ˆ
is a solution of the SHMP }:
(3.3)
O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
315
(Recall the de nition (1.3) of the Stieltjes transform .)
ˆ Thus the SHMP is determinate in the limit
point case, indeterminate in the limit
circle
case.
P
P∞
2
2
In the limit point case the series ∞
n=0 |’n (z)| and
n=0 | n (z)| 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 ∪ R̂ onto the disk ∞ (z), the open upper half-plane U onto the
interior ∞ (z) and the extended real line R̂ 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)
b(z)
c(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 ∈ R̂ 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 de ne for ∈ (0; =2) the angular region
by
= {z ∈ C − {0}: 6|arg z|6 − }:
A central result of [20] is the following theorem, valid for an arbitrary indeterminate SHMP.
(4.3)
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O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
Theorem 4.1. For arbitrary xed numbers and ”; 0¡¡=2; ”¿0; there exists a constant
M (; ”) such that
1
|f(z)|6M (; ”)exp ” |z| +
|z|
for all z∈
;
(4.4)
where f is any of the functions ; ;
; ; 1=.
The main purpose of this paper is to strenghten this result in the situation where the corresponding
SSMP is solvable. We de ne for ∈ (0; ) the angular region by
= {z ∈ C − {0}: |arg z|6 − }:
(4.5)
We shall prove the following result.
Theorem 4.2. Assume that the SSMP associated with the strong moment sequence is solvable.
Then for arbitrary xed numbers and ”; 0¡¡; ”¿0; there exists a constant H (; ”) such that
|f(z)|6H (; ”)e
1
” |z|+| |
z
for all z ∈
;
(4.6)
where f is any of the functions ; ;
; ; 1=.
Proof. Set z = Rei and consider the function hR () = |z − |2 where is some xed number in R.
We nd that hR () = R2 − 2R cos + 2 . Thus hR () is increasing with increasing || when ∈ R+ ,
decreasing with increasing || when ∈ R− . Choose some ∈ (0; =2), and consider the function hR
for some ∈ R− , e.g. = re−i . We have
hR (0) = (R + r)2 ;
Thus for z ∈
hR () = r 2 + R2 + 2rR cos :
(4.7)
−
,
(r + R)2
hR () hR (0)
6
=
hR () hR () (r + R)2 − 2rR(1 − cos )
(4.8)
hR ()
1
:
6
hR () cos
(4.9)
and since 2rR6(r + R)2 =2 we get
This means that
|z − |6 √
1
|Rei − | for z ∈
cos
−
(4.10)
when ∈ R− .
Let fn denote one of the functions n ; n ;
n ; n . We know from Proposition 2.2 that fn has at
most one negative zero. By using (4.10) and the fact that |z − | is increasing as a function of ||
for ∈ R+ , we nd that
1
fn (Rei )
|fn (z)|6 √
cos
for z ∈
−
:
(4.11)
O. Njastad / Journal of Computational and Applied Mathematics 99 (1998) 311–318
Letting n tend to in nity and combining the result with (4.4) for |arg z| = we get for all z ∈
|f(z)|6 √
1
1
M (; ”)e”(R+(1=R)) = √
M (; ”)e”(|z|+(1=|z|)) :
cos
cos
317
−
:
(4.12)
We conclude from (4.12) and Theorem 4.1 that
|f(z)|6 √
1
M (; ”)e”(|z|+(1=|z|)
cos
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 de nition (3.4) that 1=(Rei ) is increasing as a
function of ||. Combining this with (4.4) for |arg z| = we get for all z ∈ −
:
1
1
6
6M (; ”)e”(R+(1=R)) = M (; ”)e”(|z|+(1=|z|)) :
(z) (Rei )
(4.14)
We conclude from (4.14) and Theorem 4.1 that
1
6M (; ”)e”(|z|+1=|z|)
(z)
for z ∈
:
(4.15)
The desired result now follows from (4.13) and (4.15).
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