View source for Strong Stieltjes moment problem

The strong Stieltjes moment problem for a given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s1305901.png" /> of real numbers is concerned with finding real-valued, bounded, monotone non-decreasing functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s1305902.png" /> with infinitely many points of increase for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s1305903.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s1305904.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

This problem, which generalizes the classical Stieltjes moment problem (where the given sequence is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s1305905.png" />; cf. also [[Krein condition|Krein condition]]), was first studied by W.B. Jones, W.J. Thron and H. Waadeland [[#References|[a1]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s1305906.png" /> be the complex linear space spanned by the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s1305907.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s1305908.png" />, and define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s1305909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059012.png" />. An element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059013.png" /> is called a Laurent polynomial. For a given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059014.png" />, a necessary and sufficient condition for the strong Stieltjes moment problem to be solvable is that the [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059015.png" /> defined on the base elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059017.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

is positive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059019.png" />, i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059020.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059024.png" />. An equivalent condition is that if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059027.png" />, are the Hankel determinants associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059028.png" /> (cf. also [[Hankel matrix|Hankel matrix]]), then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059029.png" /></td> </tr></table>

Orthogonal Laurent polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059030.png" /> may be defined with respect to the [[Inner product|inner product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059031.png" /> and are given by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059033.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059035.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059036.png" />. Corresponding associated orthogonal Laurent polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059037.png" /> are defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

The rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059039.png" /> are the convergents of the positive T-fraction [[#References|[a3]]],

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059041.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059042.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059043.png" /></td> </tr></table>

which corresponds to the formal pair of power series,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>

The T-fraction is equivalent to the [[Continued fraction|continued fraction]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059047.png" /></td> </tr></table>

The following result may then be proved [[#References|[a1]]]: The solution of the strong Stieltjes moment problem (a1) is unique if and only if at least one of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059049.png" /> diverges, and then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059051.png" /> is this unique solution.The convergence is uniform on every compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059052.png" />.

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 [[#References|[a4]]].

A classic example of a strong Stieltjes moment problem is the log-normal distribution,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059053.png" /></td> </tr></table>

(Cf. also [[Normal distribution|Normal distribution]].) The corresponding sequence of moments is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059054.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059055.png" /></td> </tr></table>

and the strong Stieltjes moment problem in this case is indeterminate [[#References|[a5]]]. The moments corresponding to the log-normal distribution are related to a subclass of strong Stieltjes moment problems where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130590/s13059056.png" /></td> </tr></table>

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 [[#References|[a6]]]. Other subclasses have been investigated in [[#References|[a7]]].

Cf. also [[Moment problem|Moment problem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.B. Jones,   O. Njåstad,   W.J. Thron,   "A strong Stieltjes moment problem"  ''Trans. Amer. Math. Soc.'' , '''261'''  (1980)  pp. 503–528</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.B. Jones,   O. Njåstad,   W.J. Thron,   "Continued fractions and strong Hamburger moment problems"  ''Proc. London Math. Soc.'' , '''47'''  (1983)  pp. 105–123</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.B. Jones,   W.J. Thron,   "Continued fractions: Analytic theory and applications" , ''Encycl. Math. Appl.'' , '''11''' , Addison-Wesley  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O. Njåstad,   "Solutions of the strong Stieltjes moment problem"  ''Meth. Appl. Anal.'' , '''2'''  (1995)  pp. 320–347</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.K. Common,   J. McCabe,   "The symmetric strong moment problem"  ''J. Comput. Appl. Math.'' , '''67'''  (1996)  pp. 327–341</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR></table>