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