# Moment problem

One of the interpolation problems in the real or complex domain.

The first precise formulation of the original version of the moment problem in the real domain is due to T.J. Stieltjes (1894). He proposed and solved the following problem in connection with the study of continued fractions (cf. Continued fraction): Given a sequence of real numbers $\{ \mu _ {n} \}$, $n = 0 , 1 \dots$ determine a bounded and non-decreasing function $\psi ( x)$ on $[ 0 , + \infty )$ such that

$$\tag{1 } \int\limits _ { 0 } ^ \infty x ^ {n} d \psi = \mu _ {n} ,\ \ n = 0 , 1 , . . . .$$

As in every interpolation problem, the solution of (1) consists of two parts.

### Problem A.

Let $\mathfrak M$ be the set of all sequences of real numbers $\{ \mu _ {n} \}$ for which the infinite system of equations (1) has at least one solution $\psi$ with the above properties; determine necessary and sufficient (constructive) conditions which must be satisfied by the numbers $\mu _ {n}$, $n = 0 , 1 \dots$ in order that $\{ \mu _ {n} \} \in \mathfrak M$.

### Problem B.

Determine the set of all solutions in the class of bounded non-decreasing functions $\psi$ on $[ 0 , + \infty )$ satisfying the infinite system (1) for given $\mu _ {n}$, $n = 0 , 1 \dots$ $\{ \mu _ {n} \} \in \mathfrak M$.

The left-hand sides of (1) were called "momentmoments" by Stieltjes. He borrowed the terminology from mechanics. If $d \psi ( x)$ is interpreted as the mass on $[ x , x + dx ]$, then the integral $\int _ {0} ^ {X} d \psi ( t)$ is the mass on $[ 0 , X ]$. The integrals (1) for $n = 1$ and $n = 2$ are then, respectively, the first (static) and second (inertial) moments with respect to the origin $x = 0$ of the total mass $\int _ {0} ^ \infty d \psi ( x)$( this corresponds to $n = 0$ in (1)) on $[ 0 , \infty )$. Generalizing this idea, Stieltjes called the integral

$$\int\limits _ { 0 } ^ \infty x ^ {n} d \psi ( x)$$

the moment of order $n$( relative to $x = 0$) of the given mass $\int _ {0} ^ \infty d \psi ( x)$ with $\psi$ as distribution on $[ 0 , + \infty )$.

Stieltjes related the solution of the moment problem in the following way to the "natural" continued fraction associated with the integral

$$\tag{2 } I ( z , \psi ) = \ \int\limits _ { 0 } ^ \infty \frac{d \psi ( x) }{z + x } \sim \frac{\mu _ {0} }{z} - \frac{\mu _ {1} }{z ^ {2} } + \frac{\mu _ {2} }{z ^ {3} } - \frac{\mu _ {3} }{z ^ {4} } + \dots ,$$

more precisely, to the formal series

$$\sum _ { n= } 0 ^ \infty ( - 1 ) ^ {n} \frac{\mu _ {n} }{z ^ {n+} 1 } .$$

Corresponding to the integral $I ( z , \psi )$ there is a continued fraction:

$$\tag{3 } I ( z , \psi ) \sim \ \frac{1 \mid }{\mid a _ {1} z } + \frac{1 \mid }{\mid a _ {2} } + \frac{1 \mid }{\mid a _ {3} z } + \frac{1 \mid }{\mid a _ {4} } + \dots ,$$

and also a "closely related" continued fraction

$$\tag{4 } I ( z , \psi ) \sim \ \frac{\lambda _ {1} \mid }{\mid z + c _ {1} } - \frac{\lambda _ {2} \mid }{\mid z + c _ {2} } - \frac{\lambda _ {3} \mid }{\mid z + c _ {3} } - \dots .$$

The continued fraction (4) is obtained from (3) by "reductions" of the form

$$z - \frac{\alpha \mid }{\mid 1 } - \frac{\beta \mid }{\mid z - \gamma } = z - \alpha - \frac{\alpha \beta }{z - ( \beta + \gamma ) } .$$

Making use of the theory of continued fractions, Stieltjes proved that in a certain sense a necessary and sufficient condition for the solvability of (1) (which is equivalent to $\{ \mu _ {n} \} \in \mathfrak M$) is the positivity of all $a _ {n}$ in (3), which, in turn, is a consequence of the positivity of $\lambda _ {n}$ and $c _ {n}$ in (4). In terms of $\mu _ {n}$ these conditions are equivalent to the positivity of the determinants

$$\Delta = \ \mathop{\rm det} \| \mu _ {i+} j \| _ {i , j = 0 } ^ {n}$$

and

$$\Delta _ {n} ^ {(} 1) = \ \mathop{\rm det} \| \mu _ {i + j + 1 } \| _ {i , j = 0 } ^ {n} .$$

The moment problem (1) is called well-posed or determined for a given sequence $\{ \mu _ {n} \}$, $\{ \mu _ {n} \} \in \mathfrak M$, if the system (1) has a unique solution $\psi$. On the other hand, it has been shown that if the system (1) has more than one solution for a given $\mu _ {n}$, $\{ \mu _ {n} \} \in \mathfrak M$, then it has an infinite number of solutions.

Example: the two functions

$$\psi _ {1} ( x) = \ \int\limits _ { 0 } ^ { x } t ^ {- \mathop{\rm ln} t } d t$$

and

$$\psi _ {2} ( x) = \ \int\limits _ { 0 } ^ { x } t ^ {- \mathop{\rm ln} t } [ 1 - \theta \sin ( \pi \mathop{\rm ln} t ) ] d t ,\ \ \theta \in [ 0 , 1 ] ,$$

have the same moments

$$\int\limits _ { 0 } ^ \infty x ^ {n} d \psi _ {1} ( x) = \ \int\limits _ { 0 } ^ \infty x ^ {n} d \psi _ {2} ( x)$$

for all $n = 0 , 1 ,\dots$.

Stieltjes effectively constructed certain solutions of (1), which, of course, all coincide in a well-known sense if (1) is well-posed. When the moment problem (1) is ill-posed or undetermined, the Stieltjes solutions have a number of extremal properties. Stieltjes subsequently showed that (1) is well-posed or ill-posed depending on the convergence or divergence of the continued fraction (3) (which is equivalent to the divergence or convergence of the series $\sum _ {n=} 0 ^ \infty a _ {n}$). Here the fraction (3) may be convergent to $I ( z , \psi )$, whereas the series

$$\sum _ { n= } 0 ^ \infty \frac{( - 1 ) ^ {n} \mu _ {n} }{z ^ {n+} 1 }$$

may, at the same time, diverge for all $z \in \mathbf C$.

Preceding the work of Stieltjes , the moment problem in the real domain was considered in a less general and less precise formulation; such as, for example, in a series of papers by P.L. Chebyshev  and A.A. Markov . They mainly investigated the following problem: Give a description of the properties of a class $U$ of functions defined on $( - \infty , + \infty )$ such that the relations

$$p ( x) \in U$$

and

$$\tag{5 } \int\limits _ {- \infty } ^ { {+ } \infty } x ^ {n} p ( x) d x = \ \int\limits _ {- \infty } ^ { {+ } \infty } x ^ {n } e ^ {- x ^ {2} } d x ,\ n = 0 , 1 \dots$$

$$p ( x) = e ^ {- x ^ {2} } .$$

In other words, the question here concerns a maximally complete and constructive characterization of the uniqueness class $U$ of the interpolation problem (5). The solution of the moment problem (5) plays a major role in probability theory and mathematical statistics. Also of major significance are the polynomials $\omega _ {n} ( x)$, the dominators of the successive approximations (that is, the approximants) of the continued fraction (4). The study of the properties of the polynomials $\{ \omega _ {n} ( x) \}$ later initiated a broad field of research into the theory of orthogonal polynomials.

H. Hamburger (1920) generalized the moment problem (1) to the case of the whole real line $\mathbf R = ( - \infty , + \infty )$. Here the consideration of negative values of $x$ introduced a number of peculiarities and was non-trivial. Hamburger, making essential use of Helly's selection principle (cf. Helly theorem), aimed at obtaining necessary and sufficient conditions for the solvability of the system

$$\tag{6 } \int\limits _ {- \infty } ^ { {+ } \infty } x ^ {n} d \psi ( x) = \mu _ {n} ,\ \ n = 0 , 1 \dots$$

thereby completely solving the problem of convergence of the continued fractions (3) and (4) generated by (6). The union of problems $A$ and $B$ in relation to (6) is called the moment problem of equation (6). Hamburger obtained a criterion for the existence of a unique solution of the moment problem for (6). In this connection, the moment problem for (6) may be ill-posed, whereas at the same time the corresponding moment problem (1) (with the same $\mu _ {n}$) may be well-posed (have a unique solution). R. Nevanlinna (1922) gave a solution to the moment problem (6) using the integrals

$$I ( z , \psi ) = \ \int\limits _ {- \infty } ^ { {+ } \infty } \frac{d \psi ( x) }{z - x } ,\ \ x \in \mathbf C \setminus \mathbf R ,$$

and studied properties of these solutions. He made an important observation about the so-called "extremal solution" of the moment problem (6).

M. Riesz (1921) obtained solutions of the moment problem (6) based on the theory of quasi-orthogonal polynomials. These consist of linear combinations of the form $A _ {n} \omega _ {n} ( x) + A _ {n-} 1 \omega _ {n-} 1 ( x)$, where $A _ {k}$ are constants and $\omega _ {k} ( x)$ is the dominator of the $k$- th approximant of the continued fraction (4) associated with (6). He observed a close connection between the solutions of the moment problem (6) and the validity of Parseval's formula for the system of orthogonal polynomials $\{ \omega _ {k} ( x) \}$. T. Carleman (1923–1926) established connections between the moment problem (6), the theory of quasi-analytic functions and the theory of quadratic forms in a countable set of variables. He also obtained the most general criterion for the well-posedness of the moment problem (6). F. Hausdorff (1923) obtained a criterion for the solvability of the moment problem (6) $( \iff \{ \mu _ {n} \} \in \mathfrak M )$ under the condition that the function $\psi ( x)$ in (6) is a constant outside a given interval. He effectively constructed the solution $\psi ( x)$ of (6) (which, under the assumption given above, is always unique); this provides an opportunity to obtain criteria for additional properties of solutions $\psi ( x)$ of (6) (continuity, differentiability, etc.). Carleman and subsequently M.H. Stone (1932) fully investigated (6) based on results in the theory of Jacobi quadratic forms and the theory of singular integral equations. E.K. Haviland (1935) and H. Cramér (1937) extended Riesz's theory of (6) to the multi-dimensional case.

Numerous different generalizations of the moment problem have also been considered. Mainly these are variants (or a combination of variants) of the following two themes.

Replacement of the powers $x ^ {n}$ in the integrals (6) by "moment" sequences of functions $\{ \phi _ {n} ( x) \}$ of another form, and replacement of the left-hand sides of (6) by other kinds of integrals (for example, the case when $d \psi ( x)$ is replaced by $\phi ( x) d x$, where $\phi ( x) \in L _ {p}$, $p \geq 1$, has been studied) or even by operators acting in abstract spaces.

Thus, with respect to the first theme, one has the so-called trigonometric moment problem, which is the following: Given an infinite sequence of numbers $\{ c _ {n} \} _ {n = - \infty } ^ \infty$, determine a function $\psi ( x)$, non-decreasing on $[ - \pi , \pi ]$, satisfying

$$\tag{7 } \frac{1}{2 \pi } \int\limits _ {- \pi } ^ \pi e ^ {inx} d \psi ( x) = c _ {n} ,\ \ n = 0 , \pm 1 \dots$$

that is, solve problems $A$ and $B$ for the system (7).

Precise formulations of certain results concerning the theory of moment problems in the real domain are given below. Let $\mathbf R ^ {n}$ be the $n$- dimensional Euclidean space. A set function $\Phi ( e)$, defined on the family ${\mathcal B}$ of all Borel sets in $\mathbf R ^ {n}$, is called a distribution if $\Phi ( e) \geq 0$ for all $e \in {\mathcal B}$ and if

$$\sum _ { i= } 1 ^ \infty \Phi ( e _ {i} ) = \Phi \left ( \sum _ { i= } 1 ^ \infty e _ {i} \right )$$

whenever $e _ {i} \cap e _ {j} = \emptyset$, $i \neq j$, where $e _ {i} \in {\mathcal B}$ for all $i , j = 1 , 2 , . . .$.

The spectrum $\sigma ( \Phi )$ of a distribution $\Phi$ is the set of all points $x = ( x _ {1} \dots x _ {n} ) \in \mathbf R ^ {n}$ such that $\Phi ( G) > 0$ for an arbitrary open set $G \subset \mathbf R ^ {n}$ containing $x$. Let

$$\tag{8 } \{ \mu _ {i _ {1} \dots i _ {n} } \} ,\ \ i _ {1} \dots i _ {n} = 0 , 1 \dots$$

be an $n$- fold infinite sequence of real numbers. The question is: What are necessary and sufficient conditions to be satisfied by the numbers (8) in order that there is a distribution $\Phi$, with spectrum $\sigma ( \Phi )$ contained in a given closed set $F$, which is a solution of the system

$$\tag{9 } \int\limits _ {\mathbf R ^ {n} } t _ {1} ^ {i _ {1} } \dots t _ {n} ^ {i _ {n} } d \Phi = \mu _ {i _ {1} \dots i _ {n} } ,\ \ i _ {1} \dots i _ {n} = 0 , 1 ,\dots$$

(problem $A$ for (9)). Problem $B$ for (9) is formulated similarly. The union of problems $A$ and $B$ for (9) is called the $F$- moment problem. The $F$- moment problem is well-posed if its solution is in some way unique. Otherwise the $F$- moment problem (9) is called ill-posed.

### Theorem.

A necessary and sufficient condition that the $F$- moment problem (9) has a solution in $\mathbf R ^ {2}$ is that the condition

$$\sum a _ {i} b _ {j} \mu _ {ij} \geq 0$$

holds for any polynomial

$$P ( u , v ) = \sum a _ {i} b _ {j} u ^ {i} v ^ {j}$$

taking non-negative values for all $( u , v ) \in F$.

This theorem is the basis for obtaining solvability conditions (that is, for the solution of problem $A$) for different versions of (9). Here are some of them.

### Theorem 1.

In order that the moment problem (6) (with $F = \mathbf R$) have a solution it is necessary that

$$\Delta _ {n} = \mathop{\rm det} \| \mu _ {i+} j \| _ {i,j=} 0 ^ {n} \geq 0 ,\ \ n = 0 , 1 ,\dots .$$

For the existence of a solution to the moment problem (6) having a spectrum which is not a finite number of points, it is necessary and sufficient that

$$\Delta _ {n} > 0 ,\ \ n = 0 , 1 ,\dots .$$

For the existence of a solution to the moment problem (6) having a spectrum consisting of precisely $k + 1$ different points, it is necessary and sufficient that

$$\Delta _ {0} \dots \Delta _ {k} > 0 ,\ \ \Delta _ {k+} 1 = \Delta _ {k+} 2 = \dots = 0 .$$

In the latter case the moment problem (6) is always well-posed.

### Theorem 2.

In order that the moment problem (1) (with $F = [ 0 , \infty )$) is solvable it is necessary that

$$\Delta _ {n} = \mathop{\rm det} \| \mu _ {i+} j \| _ {i,j=} 0 ^ {n} \geq 0$$

and

$$\Delta _ {n} ^ {(} 1) = \ \mathop{\rm det} \| \mu _ {i+} j+ 1 \| _ {i,j=} 0 ^ {n} \geq 0 ,\ \ n = 0 , 1 ,\dots .$$

For the existence of a solution to the moment problem (1) having a spectrum which is not a finite number of points, it is necessary and sufficient that

$$\Delta _ {n} > 0 \ \textrm{ and } \ \ \Delta _ {n} ^ {(} 1) > 0 ,\ \ n = 0 , 1 ,\dots .$$

Necessary and sufficient conditions have also been obtained for the existence of a solution to the moment problem (1) having a spectrum $\sigma ( \Phi )$ consisting of precisely $k + 1$ points different from $x = 0$. The conditions are similar to those given in the final part of Theorem 1.

### Theorem 3.

A necessary and sufficient condition that the Hausdorff moment problem in $\mathbf R$,

$$\int\limits _ { 0 } ^ { 1 } x ^ {n} d \Phi = \mu _ {n} ,\ \ n = 0 , 1 \dots \ \ F = [ 0 , 1 ] ,$$

has a solution, is that $\Delta ^ {k} \mu _ {v} \geq 0$ for all $k , v = 0 , 1 ,\dots$( here $\Delta ^ {k}$ denotes the $k$- th difference operator).

### Theorem 4.

A necessary and sufficient condition that the Hausdorff moment problem in $\mathbf R ^ {2}$,

$$\int\limits _ { 0 } ^ { 1 } \int\limits _ { 0 } ^ { 1 } u ^ {i} v ^ {j} d \Phi = \mu _ {ij} ,\ \ i , j = 0 , 1 \dots \ \ F = [ 0 , 1 ] \times [ 0 , 1 ] ,$$

has a solution, is that

$$\Delta _ {1} ^ {n} \Delta _ {2} ^ {m} \mu _ {ij} \geq 0 ,\ \ n , m , i , j = 0 , 1 ,\dots .$$

### Theorem 5.

The moment problem (6) is well-posed if

$$\tag{10 } \sum _ { n= } 0 ^ \infty \frac{1}{\mu _ {2n} ^ {1/2n} } = + \infty .$$

Necessary and sufficient conditions are known (see, for example, ) which must be satisfied by $\mu _ {n}$ in order that the moment problem (6) (the moment problem (1)) be well-posed; however, these conditions are less simple than the sufficient condition (10) and their formulation is somewhat cumbersome.

The moment problem in the complex domain is the name of a wide class of interpolation problems described as follows. Let $D$ be an open simply-connected domain in the complex plane $\mathbf C$, $\infty \notin D$; let $A ( D)$ be the space of analytic functions in $D$ with topology defined by uniform convergence on arbitrary compact sets $K \subset D$; let $A ^ {*} ( D)$ be the space of all functions $\gamma ( z)$ analytic in a neighbourhood $V ^ \infty = V ^ \infty ( \gamma )$ of the point at infinity for which $\gamma ( \infty ) = 0$ and $\supp \gamma \subset D$( the latter is another way of saying that the set of singularities of $\gamma \in A ^ {*} ( D)$ lies in $D$). The topology in $A ^ {*} ( D)$ is defined by uniform convergence on one of the curves of the family of simple closed Jordan curves $\{ \Gamma _ \alpha \} \subset D$ having the property: For any compact set $K \subset D$ there is a $\Gamma _ {\alpha _ {0} } = \Gamma _ {\alpha _ {0} } ( K) \in \{ \Gamma _ \alpha \}$ such that $K \subset \mathop{\rm int} \Gamma _ {\alpha _ {0} } ( D)$( here $\mathop{\rm int} \Gamma _ \alpha$ denotes the open simply-connected domain with boundary $\Gamma _ {\alpha _ {0} }$ lying inside $\Gamma _ {\alpha _ {0} }$). It is well known that the spaces $A ( D)$ and $A ^ {*} ( D)$ are dual.

The moment problem in a complex domain is as follows. Given an integer $p > 1$, functions $0 \not\equiv A _ {s} ( z) \in A ( D)$, $s = 0 \dots p - 1$, a univalent function $W ( z) \in A ( D)$, and a set of $p$ sequences of complex numbers

$$\alpha _ {p} = \{ \{ a _ {ns} \} :\ n = 0 , 1 ,\dots; s = 0 \dots p - 1 \} ,$$

can one find a function $\gamma ( z) \in A ^ {*} ( D)$ for which

$$\tag{11 } \frac{1}{2 \pi } \int\limits _ \Gamma [ W ( z) ] ^ {np} A _ {s} ( z) \gamma ( z) d z = a _ {ns} ,$$

$$n = 0 , 1 ,\dots; \ s = 0 \dots p - 1 ,$$

where

$$\supp \gamma \subset \mathop{\rm int} \Gamma \subset \Gamma \subset D ?$$

In general, it is not true for every given collection $\alpha _ {p}$ that the infinite system (11) has at least one solution $\gamma ( z) \in A ^ {*} ( D)$. Therefore a collection $\alpha _ {p}$ is called $D$- admissible if there is (at least one) $\gamma ( z) \in A ^ {*} ( D)$ satisfying (11).

### Problem A.

Determine necessary and sufficient conditions (of a constructive nature) for the $D$- admissibility of a collection $\alpha _ {p}$.

### Problem B.

Let $\alpha _ {p}$ be $D$- admissible. The question is: How can one determine the complete set of functions $\gamma ( z) \in A ^ {*} ( D)$ satisfying (11) with respect to given numbers $a _ {ns}$ in the right-hand side of (11)?

The union of problems $A$ and $B$ is called a moment problem in the complex domain. Problem $B$, for the case $p = 1$ and $A _ {0} ( z) = 1$, was first treated in 1937 by A.O. Gel'fond ; he discussed whether, in principle, problem $B$ can be solved (for $p = 1$ and $A _ {0} ( z) = 1$ the system (11) always has a unique solution for $D$- admissible right-hand sides $a _ {ns}$). Numerous special cases of problems $A$ and $B$ have been investigated (see ). Using tools from the theory of boundary value problems allows one to attain (see ) a fairly complete investigation of the moment problem in the complex domain.

A domain $G \subset \mathbf C$ is called $2 \pi / p$- invariant, $G \in \mathop{\rm Inv} ( 2 \pi / p )$, if $\mathop{\rm exp} ( 2 \pi i / p) G \equiv G$.

An exhaustive solution to the moment problem in a complex domain $D$ under natural assumptions concerning the functions $A _ {s} ( z)$, $s = 0 \dots p - 1$, has been given in , when $W ( D) = G \in \mathop{\rm Inv} ( 2 \pi / p )$, as well as for a domain whose image $W ( D)$ can be imbedded in some domain $G \in \mathop{\rm Inv} ( 2 \pi / p )$. The theory of boundary value problems can be fruitfully used to obtain a complete solution by quadrature of problem $B$ for domains of the types indicated. In particular, for $p = 1$ every domain $G$ belongs to the class $\mathop{\rm Inv} ( 2 \pi / p )$. Thus necessary and sufficient conditions for the uniqueness of the solution of the system (11) have been found for domains $D$ whose $W$- images cannot be imbedded. These domains are important in applications. Here there are two essentially different cases: $0 \in W ( D)$ and $0 \notin W ( D)$( in the latter, the question of the uniqueness of the solution to (11) has been exhaustively studied on the assumption that $n = 0 , \pm 1 ,\dots$). Several versions of the moment problem (11) are possible with regard to the behaviour of the corresponding functions on $\Gamma$.

A number of well known interpolation problems reduce to the moment problem in the complex domain by means of the Borel transformation and its generalizations (see Comparison function and Borel transform), for example:

$$F ^ { ( n) } ( h n ) = a _ {n} ; \ \ F ^ { ( n) } ( \omega ^ {n} ) = a _ {n} ;$$

$$F ( \omega ^ {n} ) = a _ {n} ; \ \Delta ^ {n} F ( h n ) = a _ {n} ;$$

$$F ^ { ( n p + l _ {s} ) } ( \alpha _ {s} ) = a _ {ns} ;$$

$$n = 0 , 1 \dots \ s = 0 \dots p - 1 ,\ l _ {s} = 0 , 1 ;$$

$$\Delta ^ {2 n + l _ {s} } F ( \alpha _ {s} + 2 h n ) = a _ {ns} ,\ s = 0 , 1 ; \ l _ {s} = 0 , 1 ; \ n = 0 , 1 ,\dots .$$

In addition, many theorems on integer-valued functions reduce to very specific cases of problem $A$.

How to Cite This Entry:
Moment problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moment_problem&oldid=47881
This article was adapted from an original article by Yu.A. Kaz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article