Strong Stieltjes moment problem
The strong Stieltjes moment problem for a given sequence of real numbers is concerned with finding real-valued, bounded, monotone non-decreasing functions
with infinitely many points of increase for
such that
![]() | (a1) |
This problem, which generalizes the classical Stieltjes moment problem (where the given sequence is ; cf. also Krein condition), was first studied by W.B. Jones, W.J. Thron and H. Waadeland [a1].
Let be the complex linear space spanned by the set of functions
with
, and define
and
for
, and
. An element of
is called a Laurent polynomial. For a given sequence
, a necessary and sufficient condition for the strong Stieltjes moment problem to be solvable is that the linear operator
defined on the base elements
of
by
![]() | (a2) |
is positive on , i.e. for any
such that
for
and
, then
. An equivalent condition is that if
![]() | (a3) |
for ,
, are the Hankel determinants associated with
(cf. also Hankel matrix), then
![]() |
Orthogonal Laurent polynomials may be defined with respect to the inner product
and are given by:
![]() | (a4) |
![]() |
and
![]() | (a5) |
![]() |
and . Corresponding associated orthogonal Laurent polynomials
are defined by
![]() | (a6) |
The rational functions are the convergents of the positive T-fraction [a3],
![]() | (a7) |
![]() |
where
![]() |
![]() |
which corresponds to the formal pair of power series,
![]() | (a8) |
The T-fraction is equivalent to the continued fraction
![]() | (a9) |
where
![]() | (a10) |
![]() |
The following result may then be proved [a1]: The solution of the strong Stieltjes moment problem (a1) is unique if and only if at least one of the series ,
diverges, and then
![]() | (a11) |
where is this unique solution.The convergence is uniform on every compact subset of
.
The strong Stieltjes moment problem is said to be determinate when it has a unique solution and indeterminate otherwise. A detailed discussion of the latter case has been given in [a4].
A classic example of a strong Stieltjes moment problem is the log-normal distribution,
![]() |
(Cf. also Normal distribution.) The corresponding sequence of moments is , where
![]() |
and the strong Stieltjes moment problem in this case is indeterminate [a5]. The moments corresponding to the log-normal distribution are related to a subclass of strong Stieltjes moment problems where
![]() |
This subclass has been called strong symmetric Stieltjes moment problems by A.K. Common and J. McCabe, who studied properties of the related continued fractions [a6]. Other subclasses have been investigated in [a7].
Cf. also Moment problem.
References
[a1] | W.B. Jones, O. Njåstad, W.J. Thron, "A strong Stieltjes moment problem" Trans. Amer. Math. Soc. , 261 (1980) pp. 503–528 |
[a2] | W.B. Jones, O. Njåstad, W.J. Thron, "Continued fractions and strong Hamburger moment problems" Proc. London Math. Soc. , 47 (1983) pp. 105–123 |
[a3] | W.B. Jones, W.J. Thron, "Continued fractions: Analytic theory and applications" , Encycl. Math. Appl. , 11 , Addison-Wesley (1980) |
[a4] | O. Njåstad, "Solutions of the strong Stieltjes moment problem" Meth. Appl. Anal. , 2 (1995) pp. 320–347 |
[a5] | S.C. Cooper, W.B. Jones, W.J. Thron, "Orthogonal Laurent polynomials and continued fractions associated with log-normal distributions" J. Comput. Appl. Math. , 32 (1990) pp. 39–46 |
[a6] | A.K. Common, J. McCabe, "The symmetric strong moment problem" J. Comput. Appl. Math. , 67 (1996) pp. 327–341 |
[a7] | A. Sri Ranga, E.X.L. de Andrade, J. McCabe, "Some consequences of symmetry in strong distributions" J. Math. Anal. Appl. , 193 (1995) pp. 158–168 |
Strong Stieltjes moment problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_Stieltjes_moment_problem&oldid=14838