The moment problem – Williams College

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The moment problem

Let I  R be an interval. For a positive measure  on I the nth moment is defined as I xn d(x)
 provided the integral exists. If we suppose that (sn )n0 is a sequence of real numbers, the moment
problem on I consists of solving the following three problems:
(I) Does there exist a positive measure on I with moments (sn )n0 ?
In the affirmative,
(II) is this positive measure uniquely determined by the moments (sn )n0 ?
If this is not the case,
(III) how can one describe all positive measures on I with moments (sn )n0 ?
Without loss of generality we can always assume that s0 = 1. This is just a question of normalizing the
involved measures to be probability measures.
When  is a positive measure with moments (sn )n0 , we say that  is a solution to the moment problem.
If the solution to the moment problem is unique, the moment problem is called determinate. Otherwise
the moment problem is said to be indeterminate.
On the following pages we give an introduction to the classical moment problem on the real line with
special focus on the indeterminate case. For a more detailed discussion the reader is referred to Akhiezer
[1], Berg [3] or Shohat and Tamarkin [23].
There are three essentially different types of (closed) intervals. Either two end-points are finite, one end-
point is finite, or no end-points are finite. In the last case the interval is simply R and in the first two
cases one can think of [0, 1] and [0, ). For historical reasons the moment problem on [0, ) is called
the Stieltjes moment problem and the moment problem on R is called the Hamburger moment problem.
Moreover, the moment problem on [0, 1] is referred to as the Hausdorff moment problem.
It is elementary linear algebra to verify that a positive measure with finite support is uniquely determ-
ined by its moments. Applying the approximation theorem of Weierstrass and the Riesz representation
theorem, one can extend this result to hold for positive measures with compact support. The Hausdorff
moment problem is therefore always determinate. As regards existence, Hausdorff [13] proved in 1923 that
the moment problem has a solution on [0, 1] if and only if the sequence (sn )n0 is completely monotonic.
Stieltjes introduced the moment problem on [0, ) and solved the problems about existence and unique-
ness in his famous memoir "Recherches sur les fractions continues" from 1894-95, see [24]. The memoir
is devoted to the study of continued fractions of the form
                                                             1
                                                                                                          (1)
                                                                 1
                                             m1 z +
                                                                      1
                                                      l1 +
                                                                         1
                                                             m2 z +
                                                                      l2 +   
where mn , ln > 0 and z  C. We denote by Tn (z )/Un (z ) the nth convergent (or nth approximant) and
observe that Tn (z ) and Un (z ) are polynomials in z . To be precise, T2n (z ) and T2n-1 (z ) are polynomials
of degree n - 1 whereas U2n (z ) and U2n-1 (z ) are polynomials of degree n. Moreover,
                   T2n (0) = l1 + . . . + ln ,   U2n (0) = T2n-1 (0) = 1           and U2n-1 (0) = 0.

                                                             1
The moment sequence (sn )n0 comes in via the asymptotic expansion

                 Tn (z )   s0  s1  s3                  sn-1     1
                         =    - 2 + 3 - . . . + (-1)n-1 n + O n+1 ,                          |z |  .
                 Un (z )   z   z   z                    z     z
In this way the nth convergent uniquely determines the real numbers s0 , s1 , . . . , sn-1 . The condition
mn , ln > 0 is equivalent to assuming that

                            s0    s1 . . . sn-1                   s1    s2   . . . sn
                            s1    s2 . . . sn                     s2    s3   . . . sn+1
                             .     .         .   > 0 and           .     .           .       > 0,
                             .
                             .     .
                                   .         .
                                             .                     .
                                                                   .     .
                                                                         .           .
                                                                                     .
                           sn-1   sn . . . s2n-2                  sn sn+1 . . . s2n-1

which is necessary and sufficient for the moment problem to have a solution on [0, ) with infinite
support.
Stieltjes pointed out that one has to distinguish between two cases:
                                                                 
                                       mn + ln <  and                  mn + ln = .
                                 n=1                            n=1

In the first case  the indeterminate case  the continued fraction diverges for all z  C. However, the
even convergents and the odd convergents each have a limit as n   for z  C \ (-, 0]. The limits
are different and of the form
                                                                                          
                      T2n (z )                  d1 (t)           T2n-1 (z )                  d2 (t)
                     lim       =                         and    lim         =                       ,
                    n U2n (z )           0      z+t            n U2n-1 (z )           0      z+t

where 1 and 2 are different positive (and discrete) measures on [0, ) with moments (sn )n0 . In fact, the
polynomials T2n (z ), U2n (z ), T2n-1 (z ), U2n-1 (z ) converge uniformly on compact subsets of C as n  :

                                  lim T2n (z ) = P (z ),       lim T2n-1 (z ) = R(z ),
                                  n                            n
                                                                                                        (2)
                                  lim U2n (z ) = Q(z ),        lim U2n-1 (z ) = S (z ).
                                 n                             n

The entire functions P , Q, R, S satisfy the relation

                                       Q(z )R(z ) - P (z )S (z ) = 1,    z  C,

and admit only simple zeros which are  0. As we shall see later on, these four functions play an important
role in the description of the set of solutions to an indeterminate Stieltjes moment problem.
In the second case  the determinate case  the continued fraction converges uniformly on compact subsets
of C \ (-, 0] even though the polynomials Tn (z ) and Un (z ) diverge as n  . The limit of the nth
convergent has the form
                                                          
                                              Tn (z )        d (t)
                                         lim          =            ,
                                        n Un (z )       0    z+t
where  is a positive measure on [0, ) with moments (sn )n0 . In fact,  is the only positive measure on
[0, ) with moments (sn )n0 .
                                                                   
Hamburger continued the work of Stieltjes in the series of papers "Uber eine Erweiterung des Stieltjesschen
Momentenproblems" from 1920-21, see [12]. He was the first to treat the moment problem as a theory
of its own and considered more general continued fractions than the one in (1). The role of [0, ) in
Stieltjes' work is taken over by the real line in Hamburger's work. A key result  often referred to as
Hamburger's theorem  says that (sn )n0 is a moment sequence if and only if it is positive definite. But
besides the question about existence, Hamburger was also interested in the question about uniqueness.
To avoid confusion at this point we emphasize that if (sn )n0 is a sequence of Stieltjes moments, then one
has to distinguish between determinacy and indeterminacy in the sense of Stieltjes and in the sense of
Hamburger. Obviously, an indeterminate Stieltjes moment problem is also indeterminate in the sense of

                                                           2
Hamburger and if the solution to a determinate Hamburger moment problem is supported within [0, ),
the moment problem is also determinate in the sense of Stieltjes. But a determinate Stieltjes moment
problem can just as well be determinate as indeterminate in the sense of Hamburger. In the following we
let the words determinate and indeterminate refer to the Hamburger moment problem unless otherwise
stated.
It is desirable to be able to tell whether the moment problem is determinate or indeterminate just by
looking at the moment sequence (sn )n0 . Hamburger came up with a solution to this problem by pointing
out that the moment problem is determinate if and only if

                                                s0    s1 . . . sn-1
                                                s1    s2 . . . sn
                                                 .
                                                 .     .
                                                       .         .
                                                                 .
                                                 .     .         .
                                               sn-1   sn . . . s2n-2
                                        lim                             = 0.
                                       n       s4     s5 . . . sn+1
                                               s5     s6 . . . sn+2
                                                .
                                                .      .
                                                       .           .
                                                                   .
                                                .      .           .
                                              sn+1 sn+2 . . . s2n-2

More recently, Berg, Chen and Ismail [4] have proved that the moment problem is determinate if and
only if the smallest eigenvalue of the Hankel matrix (si+j )0i,j n tends to 0 as n  . A simpler
criterion, however, was given by Carleman in his treatise of quasi-analytic functions from 1926, see [8].
He proved that if
                                            
                                                 1
                                                
                                               2n s
                                                      = ,                                            (3)
                                           n=1     2n

then the moment problem is determinate. Carleman's criterion has the disadvantage that it only gives a
sufficient condition for the moment problem to be determinate. There are moment sequences (sn )n0 for
which the series in (3) converges although the moment problem is determinate. But Carleman's criterion
tells us that the moment problem is determinate unless the even moments tend to infinity quite rapidly.
On the other hand, we cannot conclude that the moment problem is indeterminate just because the
moment sequence increases very rapidly.
Given a positive measure  with moments (sn )n0 , the orthonormal polynomials (Pn ) are characterized
by Pn (x) being a polynomial of degree n with positive leading coefficient such that

                                       Pn (x)Pm (x) d(x) = mn ,       n, m  0.
                                   R

The polynomials (Pn ) only depend on the moment sequence (sn )n0 and they can be obtained from the
formula
                                                   s0 s1 . . . sn
                                                   s1 s2 . . . sn+1
                                           1        .   .         .
                              Pn (x) =              .
                                                    .   .
                                                        .         .
                                                                  .   ,                        (4)
                                         Dn-1 Dn
                                                  sn-1 sn . . . s2n-1
                                                    1 x . . . xn

where Dn = det (si+j )0i,j n denotes the Hankel determinant. It is well-known that (Pn ) satisfy a
three-term recurrence relation of the form

                         xPn (x) = bn Pn+1 (x) + an Pn (x) + bn-1 Pn-1 (x),       n  1,                     (5)

where an  R and bn > 0. The initial conditions are P0 (x) = 1 and P1 (x) = b1  0
                                                                                 (x - a0 ). Vice versa, if (Pn )
satisfy the above three-term recurrence relation (including the initial conditions) for some real sequences
(an ) and (bn ) with bn > 0, then it follows by Favard's theorem that there exists a positive measure  on
R such that the polynomials (Pn ) are orthonormal with respect to .

                                                        3
As can be read of from (5), the leading coefficient of Pn (x) is given by (b0 b1    bn-1 )-1 . The polynomials
pn (x) := (b0 b1    bn-1 )Pn (x) are therefore monic and they satisfy the three-term recurrence relation

                            xpn (x) = pn+1 (x) + cn pn (x) + n pn-1 (x),          n  1,                      (6)

where cn = an  R and n = b2
                          n-1 > 0.

The recurrence coefficients in (5) and (6) contain useful information about the moment problem. Carleman
proved in 1922 that the moment problem is determinate if
                                                    
                                                       1
                                                         = .                                                 (7)
                                                      b
                                                   n=0 n

This condition is clearly satisfied if the sequence (bn ) is bounded and if the sequence (an ) is bounded too,
the unique solution has compact support. Just like Carleman's condition (3), the condition (7) is only
sufficient for the moment problem to be determinate. The moment problem may be determinate even
though the series in (7) converges.
In the set-up of Stieltjes the recurrence coefficients from (5) are given by
                                  1      1     1                                 1
                          an =             +               and bn =          
                                 mn+1   ln   ln+1                     ln+1      mn+1 mn+2
                                   1 1
with the convention that a0 = m     1 l1
                                         . After a few computations we see that the moment problem is
determinate in the sense of Stieltjes if (but not only if)
                                                   
                                                       1
                                                        = .
                                                   n=0
                                                       bn

Using the concept of chain sequences, Chihara proved the following result in [10]. On the assumption that
                                                    n+1       1
                               cn   and                    L<           as n  ,
                                                   cn cn+1    4
the moment problem is determinate if
                                                                
                                                             1 + 1 - 4L
                                         lim inf   c1
                                                    n
                                                      /n
                                                           <    
                                          n                  1 - 1 - 4L
and indeterminate if the opposite (strict) inequality holds. In particular, if cn has the form

                                                   cn = fn q -n ,

where 0 < q < 1 and (fn ) is both bounded and bounded away from 0, then the moment problem is
determinate if
                                                q
                                         L<
                                             (1 + q )2
and indeterminate if the opposite (strict) inequality holds.
Just like the orthonormal polynomials (Pn ), the polynomials of the second kind (Qn ) are generated
by the three-term recurrence relation (5)  but with initial conditions Q0 (x) = 0 and Q1 (x) = 1/b0 .
Consequently, (Pn ) and (Qn ) are linearly independent solutions to (5) and together they span the solution
space. Notice that Qn (x) is a polynomial of degree n - 1 and when  is a positive measure with moments
(sn )n0 , we have
                                                  Pn (x) - Pn (y )
                                     Qn (x) =                      d(y ).
                                                R      x-y

The orthonormal polynomials (Pn ) and the polynomials of the second kind (Qn ) play a crucial role for
the moment problem. Hamburger proved that the moment problem is indeterminate if and only if
                                           
                                                 2
                                                Pn (0) + Q2
                                                          n (0) < .                                          (8)
                                          n=0


                                                            4
Actually, it is necessary and sufficient that there exists an x  R such that (8) is fulfilled with x instead
of 0. It is even necessary and sufficient that there exists a z  C \ R such that either (Pn (z )) or (Qn (z ))
belong to 2 . In any case, when the moment problem is indeterminate the series
                                                                
                                                      2                       2
                                             |Pn (z )|    and         |Qn (z )|
                                       n=0                      n=0

converge uniformly on compact subsets of C.
Hamburger pointed out that in the set-up of Stieltjes the condition (8) is equivalent to
                                         
                                              mn+1 (l1 + . . . + ln )2 < .                                (9)
                                        n=1

This simply follows from the fact that

                                      Pn (z ) = (-1)n      mn+1 /m1 U2n (-z )

and
                                                      
                                     Qn (z ) = (-1)n-1 mn+1 m1 T2n (-z ).
The condition (9) enables us to determine whether a determinate Stieltjes moment problem is determinate
or indeterminate in the sense of Hamburger.
Sometimes the natural starting point is not the orthogonal polynomials but a density w(t) with moments
(sn )n0 . In this situation Krein [14] proved that the moment problem is indeterminate if

                                              1       log w(t)
                                                               dt > -.                                   (10)
                                                  R    1 + t2

Krein's condition (10) is only sufficient and not necessary for the moment problem to be indeterminate.
We shall now take a closer look at the set of solutions to an indeterminate Hamburger moment problem.
Such a set  which we will denote by VH  is clearly convex and therefore infinite. In fact, it is infinite
dimensional. Equipped with the vague topology, VH is a compact set in which the subsets of absolutely
continuous, discrete and continuous singular solutions each are dense, see Berg and Christensen [5].
Moreover, Naimark [17] proved that  is an extreme point in VH if and only if the polynomials C[x] are
dense in L1 (R, ).
The problem about describing VH was solved by Nevanlinna in 1922 using complex function theory,
see [18]. We call a function  a Pick function if it is holomorphic in the upper half-plane Im z > 0
and Im (z )  0 for Im z > 0. By reflection in the real line any such function can be extended to a
holomorphic function in C \ R. Nevanlinna proved that VH can be parametrized by the space P of Pick
functions augmented with the point . The space P inherits the topology of the holomorphic functions
on C \ R and one can think of P  {} as a one-point compactification of P . The parametrization is
established via the homeomorphism    of P  {} onto VH given by

                                    d (t)    A(z )(z ) - C (z )
                                           =-                    ,         z  C \ R,
                                R    t-z      B (z )(z ) - D(z )

where A, B , C , D are certain entire functions defined in terms of the orthonormal polynomials (Pn ) and
the polynomials of the second kind (Qn ). More precisely, A, B , C , D are the uniform limits (on compact
subsets of C) of the polynomials

                               An (z ) = bn Qn (0)Qn+1 (z ) - Qn+1 (0)Qn (z ) ,
                               Bn (z ) = bn Qn (0)Pn+1 (z ) - Qn+1 (0)Pn (z ) ,
                                                                                                         (11)
                               Cn (z ) = bn Pn (0)Qn+1 (z ) - Pn+1 (0)Qn (z ) ,
                               Dn (z ) = bn Pn (0)Pn+1 (z ) - Pn+1 (0)Pn (z ) ,

                                                           5
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