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 , determine a bounded and non-decreasing function on such that

(1)

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

Problem A.

Let be the set of all sequences of real numbers for which the infinite system of equations (1) has at least one solution with the above properties; determine necessary and sufficient (constructive) conditions which must be satisfied by the numbers , in order that .

Problem B.

Determine the set of all solutions in the class of bounded non-decreasing functions on satisfying the infinite system (1) for given , .

The left-hand sides of (1) were called "momentmoments" by Stieltjes. He borrowed the terminology from mechanics. If is interpreted as the mass on , then the integral is the mass on . The integrals (1) for and are then, respectively, the first (static) and second (inertial) moments with respect to the origin of the total mass (this corresponds to in (1)) on . Generalizing this idea, Stieltjes called the integral

the moment of order (relative to ) of the given mass with as distribution on .

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

(2)

more precisely, to the formal series

Corresponding to the integral there is a continued fraction:

(3)

and also a "closely related" continued fraction

(4)

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

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 ) is the positivity of all in (3), which, in turn, is a consequence of the positivity of and in (4). In terms of these conditions are equivalent to the positivity of the determinants

and

The moment problem (1) is called well-posed or determined for a given sequence , , if the system (1) has a unique solution . On the other hand, it has been shown that if the system (1) has more than one solution for a given , , then it has an infinite number of solutions.

Example: the two functions

and

have the same moments

for all .

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 ). Here the fraction (3) may be convergent to , whereas the series

may, at the same time, diverge for all .

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 [2] and A.A. Markov [3]. They mainly investigated the following problem: Give a description of the properties of a class of functions defined on such that the relations

and

(5)

lead to the identity

In other words, the question here concerns a maximally complete and constructive characterization of the uniqueness class 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 , the dominators of the successive approximations (that is, the approximants) of the continued fraction (4). The study of the properties of the polynomials 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 . Here the consideration of negative values of 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

(6)

thereby completely solving the problem of convergence of the continued fractions (3) and (4) generated by (6). The union of problems and 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 ) may be well-posed (have a unique solution). R. Nevanlinna (1922) gave a solution to the moment problem (6) using the integrals

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 , where are constants and is the dominator of the -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 . 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) under the condition that the function in (6) is a constant outside a given interval. He effectively constructed the solution of (6) (which, under the assumption given above, is always unique); this provides an opportunity to obtain criteria for additional properties of solutions 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 in the integrals (6) by "moment" sequences of functions of another form, and replacement of the left-hand sides of (6) by other kinds of integrals (for example, the case when is replaced by , where , , 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 , determine a function , non-decreasing on , satisfying

(7)

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

Precise formulations of certain results concerning the theory of moment problems in the real domain are given below. Let be the -dimensional Euclidean space. A set function , defined on the family of all Borel sets in , is called a distribution if for all and if

whenever , , where for all .

The spectrum of a distribution is the set of all points such that for an arbitrary open set containing . Let

(8)

be an -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 , with spectrum contained in a given closed set , which is a solution of the system

(9)

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

Theorem.

A necessary and sufficient condition that the -moment problem (9) has a solution in is that the condition

holds for any polynomial

taking non-negative values for all .

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

Theorem 1.

In order that the moment problem (6) (with ) have a solution it is necessary that

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

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

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

Theorem 2.

In order that the moment problem (1) (with ) is solvable it is necessary that

and

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

Necessary and sufficient conditions have also been obtained for the existence of a solution to the moment problem (1) having a spectrum consisting of precisely points different from . 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 ,

has a solution, is that for all (here denotes the -th difference operator).

Theorem 4.

A necessary and sufficient condition that the Hausdorff moment problem in ,

has a solution, is that

Theorem 5.

The moment problem (6) is well-posed if

(10)

Necessary and sufficient conditions are known (see, for example, [4]) which must be satisfied by 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 be an open simply-connected domain in the complex plane , ; let be the space of analytic functions in with topology defined by uniform convergence on arbitrary compact sets ; let be the space of all functions analytic in a neighbourhood of the point at infinity for which and (the latter is another way of saying that the set of singularities of lies in ). The topology in is defined by uniform convergence on one of the curves of the family of simple closed Jordan curves having the property: For any compact set there is a such that (here denotes the open simply-connected domain with boundary lying inside ). It is well known that the spaces and are dual.

The moment problem in a complex domain is as follows. Given an integer , functions , , a univalent function , and a set of sequences of complex numbers

can one find a function for which

(11)

where

In general, it is not true for every given collection that the infinite system (11) has at least one solution . Therefore a collection is called -admissible if there is (at least one) satisfying (11).

Problem A.

Determine necessary and sufficient conditions (of a constructive nature) for the -admissibility of a collection .

Problem B.

Let be -admissible. The question is: How can one determine the complete set of functions satisfying (11) with respect to given numbers in the right-hand side of (11)?

The union of problems and is called a moment problem in the complex domain. Problem , for the case and , was first treated in 1937 by A.O. Gel'fond [6]; he discussed whether, in principle, problem can be solved (for and the system (11) always has a unique solution for -admissible right-hand sides ). Numerous special cases of problems and have been investigated (see [7]–[10]). Using tools from the theory of boundary value problems allows one to attain (see [11]–[14]) a fairly complete investigation of the moment problem in the complex domain.

A domain is called -invariant, , if .

An exhaustive solution to the moment problem in a complex domain under natural assumptions concerning the functions , , has been given in [10], when , as well as for a domain whose image can be imbedded in some domain . The theory of boundary value problems can be fruitfully used to obtain a complete solution by quadrature of problem for domains of the types indicated. In particular, for every domain belongs to the class . Thus necessary and sufficient conditions for the uniqueness of the solution of the system (11) have been found for domains whose -images cannot be imbedded. These domains are important in applications. Here there are two essentially different cases: and (in the latter, the question of the uniqueness of the solution to (11) has been exhaustively studied on the assumption that ). Several versions of the moment problem (11) are possible with regard to the behaviour of the corresponding functions on .

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:

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

References

[1a]	T.J. Stieltjes, "Recherches sur les fractions continues" Ann. Fac. Sci. Univ. Toulouse , 8 (1894) pp. 1–122
[1b]	T.J. Stieltjes, "Recherches sur les fractions continues" Ann. Fac. Sci. Univ. Toulouse , 9 (1895) pp. 1–47
[2]	P.L. Chebyshev, "Oeuvres de P.L. Tchebycheff" , 1–2 , Chelsea (1961) (Translated from Russian)
[3]	A.A. Markov, "Selected work on the theory of continued fractions and the theory of functions deviating least from zero" , Moscow-Leningrad (1948) (In Russian)
[4]	J.A. Shohat, J.D. Tamarkin, "The problem of moments" , Amer. Math. Soc. (1950)
[5]	N.I. Akhiezer, "The classical moment problem and related questions in analysis" , Hafner (1965) (Translated from Russian)
[6]	A.O. [A.O. Gel'fond] Gelfond, "Differenzenrechnung" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
[7]	R.C. Buck, "Interpolation series" Trans. Amer. Math. Soc. , 64 (1948) pp. 283–298
[8]	R.C. Buck, "Integral valued entire functions" Duke Math. J. , 15 (1948) pp. 879–891
[9]	R.C. Buck, "On admissibility of sequences and a theorem of Pólya" Comment. Mat. Helv. , 27 (1953) pp. 75–80
[10]	I.F. Lokhin, "An interpolation problem for entire functions" Mat. Sb. , 35 : 2 (1954) pp. 223–230 (In Russian)
[11]	Yu.A. Kaz'min, "On a general problem in the theory of interpolation" Soviet Math. Dokl. , 11 (1970) pp. 1357–1361 Dokl. Akad. Nauk SSSR , 194 : 6 (1970) pp. 1251–1254
[12]	Yu.A. Kaz'min, "On the moment problem in the complex domain" Soviet Math. Dokl. , 13 (1972) pp. 833–837 Dokl. Akad. Nauk SSSR , 204 : 6 (1972) pp. 1309–1312
[13]	Yu.A. Kaz'min, "The general moment problem in the complex domain. Uniqueness theorems" Soviet Math. Dokl. , 13 (1972) pp. 868–872 Dokl. Akad. Nauk SSSR , 205 : 1 (1972) pp. 19–22
[14]	L. Bieberbach, "Analytische Fortsetzung" , Springer (1955)
[15]	F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian)
[16]	M.G. Krein, A.A. Nudel'man, "The Markov moment problem and extremal problems" , Amer. Math. Soc. (1977) (Translated from Russian)

Comments

The classical moment problem is connected with a large number of fundamental theoretical and applied topics, including function theory, spectral decomposition of operators, positive definiteness, probability, approximation theory, electrical and mechanical inverse problems, prediction of stochastic processes, and the design of algorithms for signal-processing VLSI chips. A survey of some of these ramifications is given in [a23].

As is obvious from the main article above, there is an intimate connection between moment problems and continued fractions; from there it is but a very small step to the field of rational approximation and interpolation, viz. Padé and Hermite–Padé approximation (cf., e.g., Padé approximation).

The vast literature on the last mentioned subject contains many papers connected with and having effect on several types of moment problems, cf. contributions in [a3]–[a12]; also, applications in physics should be mentioned, see [a13]–[a15].

Moreover, in the last 15 years the study of different types of extensions of the moment problem in the setting of the theory of continued fractions and linear analysis has intensified; cf. [a16]–[a20].

Finally, [a1]– are important from a historical point of view.

References

[a1]	J. Grommer, "Ganze transzendente Funktionen mit lauter reellen Nullstellen" J. Reine Angew. Math. , 144 (1914) pp. 212–238
[a2a]	H. Hamburger, "Ueber eine Erweiterung des Stieltjesschen Momentenproblems I" Math. Ann. , 81 (1920) pp. 235–319
[a2b]	H. Hamburger, "Ueber eine Erweiterung des Stieltjesschen Momentenproblems II" Math. Ann. , 82 (1921) pp. 120–164
[a2c]	H. Hamburger, "Ueber eine Erweiterung des Stieltjesschen Momentenproblems III" Math. Ann. , 82 (1921) pp. 168–187
[a3]	P.R. Graves (ed.) , Padé approximants and their application (Canterbury, 1972) , Acad. Press (1973)
[a4]	E.B. Saff (ed.) R.S. Varga (ed.) , Padé and Rational Approximation (Tampa, 1976) , Acad. Press (1977)
[a5]	L. Wuytack (ed.) , Padé approximation and its applications (Antwerp, 1979) , Lect. notes in math. , 765 , Springer (1979)
[a6]	M.G. de Bruin (ed.) H. van Rossum (ed.) , Padé approximation and its applications (Amsterdam, 1980) , Lect. notes in math. , 888 , Springer (1981)
[a7]	J. Gilewicz (ed.) , Proc. 1-st French-Polish Meeting on Padé Approximation and Convergence Acceleration Techniques (Warszaw, 1981) , CPT-81/PE 1354 , CNRS (1982)
[a8]	H. Werner (ed.) H.-J. Bünger (ed.) , Padé approximation and its application (Bad Honnef, 1983) , Lect. notes in math. , 1071 , Springer (1984)
[a9]	P.R. Graves-Morris (ed.) E.B. Saff (ed.) R.S. Varga (ed.) , Rational Approximation and Interpolation (Tampa, 1983) , Lect. notes in math. , 1105 , Springer (1984)
[a10]	C. Brezinski (ed.) A. Draux (ed.) A.P. Magnus (ed.) P. Maroni (ed.) A. Ronveaux (ed.) , Polynômes Orthogonaux et Applications (Bar-le-Duc, 1984) , Lect. notes in math. , 1171 , Springer (1985)
[a11]	J. Gilewicz (ed.) M. Pindor (ed.) W. Siemasko (ed.) , Rational Approximation and its Application in Mathematics and Physics (Lańcut, 1985) , Lect. notes in math. , 1237 , Springer (1987)
[a12]	A. Cuyt (ed.) , Nonlinear numerical methods and rational approximation (Antwerp, 1987) , Reidel (1988)
[a13]	J. Antolin, A. Cruz, J. Phys. , G12 (1986) pp. 297
[a14]	C.T. Corcoran, P.W. Langhoff, "Moment-theory approximations for nonnegative spectral densities" J. Math. Phys. , 18 (1977) pp. 651–657
[a15]	P.W. Langhoff, B.J. Dalton (ed.) et al. (ed.) , Moment methods in many Fermion systems , Plenum (Forthcoming)
[a16]	W.B. Jones, W.J. Thron, H. Waadeland, "A strong Stieltjes moment problem" Trans. Amer. Math. Soc. , 261 (1980) pp. 503–528
[a17]	W.J. Thron, "Survey of continued fraction methods in solving moment problems and related topics" W.B. Jones (ed.) W.J. Thron (ed.) E.H. Waadeland (ed.) , Analytic theory of continued fractions , Lect. notes in math. , 932 , Springer (1982) pp. 4–36
[a18]	W.B. Jones, W.J. Thron, O. Njastad, "Orthogonal Laurent polynomials and the strong Hamburger moment problem" J. Math. Anal. Applic. , 98 (1984) pp. 528–554
[a19]	W.B. Jones, O. Njastad, W.J. Thron, "Continued fractions associated with the trigonometric and other strong moment problems" (To appear)
[a20]	W.B. Jones, O. Njastad, W.J. Thron, "Perron–Carathéodory continued fractions" (To appear)
[a21]	N.I. Akhiezer, M. Krein, "Some questions in the theory of moments" , Amer. Math. Soc. (1962) (Translated from Russian)
[a22]	H.J. Landau, "The classical moment problem: Hilbertian proofs" J. Funct. Anal. , 38 (1980) pp. 255–272
[a23]	H.J. Landau (ed.) , Moments in mathematics , Amer. Math. Soc. (1987) pp. 56ff
[a24]	I.P. Natanson, "Constructive theory of functions" , 1–2 , F. Ungar (1964–1965) (Translated from Russian)