International Journal of Analysis

Volume 2013 (2013), Article ID 901318, 17 pages

http://dx.doi.org/10.1155/2013/901318

Review Article

New Results on Markov Moment Problem

Octav Olteanu

Department of Mathematics-Informatics, University Politehnica of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania

Received 14 September 2012; Revised 17 November 2012; Accepted 19 November 2012

Academic Editor: Remi Léandre

Copyright © 2013 Octav Olteanu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The present work deals with the existence of the solutions of some Markov moment problems. Necessary conditions, as well as necessary and sufficient conditions, are discussed. One recalls the background containing applications of extension results of linear operators with two constraints to the moment problem and approximation by polynomials on unbounded closed finite-dimensional subsets. Two domain spaces are considered: spaces of absolute integrable functions and spaces of analytic functions. Operator valued moment problems are solved in the latter case. In this paper, there is a section that contains new results, making the connection to some other topics: bang-bang principle, truncated moment problem, weak compactness, and convergence. Finally, a general independent statement with respect to polynomials is discussed.

1. Introduction and Known Results

We recall that the classical formulation of the moment problem, under the terms of T. Stieltjes, given in 1894-1895 (see [1] for details), finds the repartition of the positive mass on the semiaxis ), if the moments of arbitrary orders   are given. Precisely, in the Stieltjes moment problem, a sequence of real numbers is given, and one looks for a nondecreasing real function () that verifies the moment conditions:

This is a one-dimensional moment problem, on an unbounded interval. The existence, the uniqueness, and the construction of the solution are considered. The present paper concerns firstly the existence problem. If the interval is replaced by a subset of , , we have a multidimensional moment problem. If the sequence of the real numbers moments becomes a sequence of operators, we have an operator-valued moment problem. Most of the problems appearing in applications require not only positive, but also an upper constraint on the solution. This is the Markov moment problem.

Applying extension Hahn-Banach type results in existence questions concerning the moment problem is a well-known technique [1–13]. One of the most useful results is the lemma of the majorizing subspace (see [14, Section 5.1.2] for the proof of the lattice version of this lemma; see also [15]). It says that if is a linear positive operator on a subspace of the ordered vector space , the target space being an order-complete vector lattice ,   and for each , there is , , then has a linear positive extension . Another geometric remark is that in the real case, the sublinear functional from Hahn-Banach theorem can be replaced by a convex one. The theorem remains valid when the convex dominating functional is defined on a convex subset with some properties with respect to the subspace (e.g., ri), where ri is the relative interior of . The problem is to find necessary and sufficient conditions for the existence of a solution, preserving sandwich conditions. Here we recall an answer published in 1978 [16], without loosing convexity, but strongly generalizing the classical result. The first detailed proof was published in [17]. A proof in terms of the moment problem was published in [12]. Parts of these generalizations of the Hahn-Banach principle are involved in the present work too. Throughout this first part, will be a real vector space, an order-complete vector lattice, a convex subsets, a concave operator, a convex operator, a vector subspace, a linear operator.

Theorem 1. Assume that , .

The following assertions are equivalent:

(a)there is a linear extension of the operator such that

(b)there are convex and concave operators such that for all

One has:

The following result related to the theorem of H. Bauer (see [15, Section 5.4]) is deduced.

Theorem 2. Let be a preordered vector space with its positive cone a convex operator, a vector subspace, and a linear positive operator. The following assertions are equivalent:

(a)there is a linear positive extension of such that

(b) for all such that .

Now, we can deduce the main results on the abstract moment problem [11].

Theorem 3. Let , ,  be as in Theorem 2, , given families. The following statements are equivalent:

(a)there is a linear positive operator such that

(b)for any finite subset and any , one has

A clearer sandwich-moment problem variant is the following one.

Theorem 4. Let ,   ,   be as in Theorem 3 and , two linear operators. The following statements are equivalent:

(a)there is a linear operator such that

(b)for any finite subset and any , one has:

Some of the results of this work are applications of the theorems stated previously. Most of the results were published in the last ten years, a few of them being new (Section 4).

For uniqueness of the solution, see [1, 3, 4, 18–22]. Several other interesting related results are contained in [20, 23] (application of fixed-point principle, iterative methods, and complex functions methods) [8, 24], (construction of a solution) [13, 25]. For stable algorithms related to the finite Markov moment problem, see [20, 24]. Similar results are deduced in Section 4 of the present work, by using other methods: bang-bang principle, weak compactness, weak approximation [26], characteristic functions as extreme “points,” Gram-Schmidt procedure, and cell-decomposition. These methods seem to work in arbitrary several dimensions.

2. Approximation on Unbounded Subsets and Moment Problems in Spaces of “Even” Functions

The results of this section are using those published in [9, 10]. We recall the following recent approximation lemmas concerning the approximation on closed unbounded finite-dimensional subsets.

Lemma 5 (see [9] and [10, Lemma 1.3(d)]). If is a nonnegative continuous function with compact support, then there exists a sequence of positive polynomials on , such that

uniformly on compact subsets of .

The idea of the proof is to add the point and to apply the Stone-Weierstrass theorem to the subalgebra generated by the functions ,  . Then, one uses for each such -function suitable majorizing or minorizing partial polynomial sums, as well as the relation

Note that [10, Lemma 1.4] asserts the density of positive polynomials in , for any closed subset of a finite-dimensional space, , being a positive regular Borel M-determinate measure. Here is the exact formulation of this result.

Lemma 6 (see [10, Lemma 1.4]). Let be an arbitrary closed subset and an M-determinate positive regular Borel measure on , with finite moments of all natural orders. Then, for any such that for some polynomial , there exists a sequence ,, in . In particular, one has:

is dense in , and is dense in ,

Recall that is the subspace of all polynomial functions on .

Lemma 7. For any simple function of the form

being intervals, there exists a sequence of polynomial

where is an M-determinate positive regular Borel measure with finite moments of all natural orders. If is extended to an even function in , then the polynomials are restrictions to the positive semiaxes of even polynomials in .

Note that in the preceding Lemmas 5–7, the sequence is not monotone, but its elements are positive on their domain. The proof of Lemma 6 uses Hahn-Banach theorem, while that of Lemma 5 is more “constructive,” the latter being a statement independent with respect to the notion of a measure.

Remark 8. The general form of positive even polynomials on the real axes is

We start by proving some results on the one-dimensional moment problem.

Theorem 9 (see [10, Theorem 2.1]). Let be a positive determinate Borel regular measure on with finite moments of all orders, such that for any odd function . Let be a given sequence of real numbers.

The following statements are equivalent:

(a)there exists a Borel function such that

(b)for any finite subset and any , one has

Proof. The hypothesis (a) implies that (b) is obvious because of the qualities of .

(b)(a). Since vanishes on the subspace of odd functions, the same property is valid for . The problem can be reduced to a corresponding one on the positive semiaxis . Using the form of the positive polynomials in [1], the hypothesis (b) yields

Since is a majorizing subspace in

there exists a positive linear extension of . By Haviland-Riesz representation theorem, there exists a representing positive regular Borel measure on such that

Applying Haviland-Riesz theorem once more, we infer the existence of a positive regular Borel measure on , such that

Since is M-determinate, application of Lemma 7 and the last relation lead to

Borel subset. In particular, is absolutely continuous with respect to , hence there exists such that on simple functions. From [28, Theorem 1.40], we infer that -a.e. Since was a positive regular Borel measure, we can find a Borel function such that -a.e. The previous relations lead to

This concludes the proof.

Corollary 10. Let and . The following statements are equivalent:

(a)there exists a Borel function , such that

(b)for any finite subset and any , one has

Proof. One applies Theorem 9 to , as a measure on .

Due to Carleman criterion, is M-determinate. A straightforward computation shows that

The conclusion follows.

Using a quite different proof with respect to that of Theorem 9, we proved the following “abstract operatorial version.” Let be a positive measure as in Theorem 9, vanishing on the subspace of odd functions from . Let be a Banach lattice, which is order complete. Let , , , . Then, the following theorem holds.

Theorem 11 (see [10, Theorem 2.1]). Let be a linear continuous positive operator vanishing on the subspace of odd functions and .

The following statements are equivalent:

(a)there exists a linear operator such that

(b)for any finite subset and any , one has

In the sequel, will be the order-complete vector lattice, which is also a commutative Banach algebra of selfadjoint operators [6, 14]. Let be a Hilbert space a self-adjoint operator from into ,

Let be the space of all continuous functions applying into , such that is majorized by a polynomial on . Denote by the subspace of even functions, and will be the spectrum of .

Theorem 12. Let . The following statements are equivalent:

(a)there exists such that

(b)for any finite subset and any , one has

Proof. (b)(a). Let

Then, the assumptions of (b) become

Since is a majorizing subspace in , has a positive linear extension

Define

Then, is a linear positive extension of to the space . The first two properties of the solution hold. Using Urysohn’s lemma, for any , there is a sequence

Applications of the one-dimensional variant of Lemma 5 lead to the existence of a sequence of polynomial functions such that

These arguments and the positivity of yield

The proof is finished.

The last result of this section concerns the multidimensional moment problems with solutions vanishing on some subspaces. We prove the result only for the bidimensional case. Let

If is a Hilbert space and are two bounded positive commuting selfadjoint operators, one defines

Then, endowed with the usual order relation on selfadjoint operators is an order-complete vector lattice and a commutative Banach algebra.

Theorem 13. Let be a given sequence. The following statements are equivalent:

(a)there exists such that

(b)for all finite subsets, for all , and for all , one has

Proof. Since (a)(b) is obvious, we have to prove the implication (b)(a). By Stone-Weierstrass theorem, we know that the vector subspace generated by the functions

is a dense subalgebra of . Hence, for any , there exists a sequence of elements from , which converges uniformly to on , such that in . For any function from , we consider its extension to which vanishes outside , and then we extend again this function to , such that , for all . If we write

where are extended to such that to vanish outside , and to such that to be even functions, we infer that are even bounded measurable nonnegative functions on with compact support. An application of Luzin’s theorem and of the one-dimensional version of Lemma 5 [10, lemma 1.3(b)] leads to uniform approximation of on , by even positive polynomials on . Using the representation of these polynomials given in Remark 8, (b) is rewritten as

In particular, is a linear positive operator on the subspace generated by the products of even polynomials in each variable, which is a majorizing subspace in . This yields the existence of a positive extension of to the whole space . The positivity of and the uniform convergence of to on lead to

The last relation (a) follows too. This concludes the proof.

3. Applying the Solutions of the Abstract Moment Problems to Some Classical Moment Problems

The results of this section follow the papers in [7, 8]. In the next theorem, will be the space of all absolutely convergent power series:

One assumes that the elements of the space are continuous in the closed disk . Let be a Hilbert space, a selfadjoint operator acting on , and the order-complete vector lattice and commutative Banach algebra of selfadjoint operators defined in Section 2 (see also [6, 14]). We denote that , , .

Theorem 14. Let , , , , be as previously mentioned. Assume that and let , . Let and . The following statements are equivalent:

(a)there exists a linear operator applying into such that

(b)one has

Proof. (a)(b) is obvious, since from we derive

(b)(a). One applies Theorem 4 to . We have

The preceding relations yield

Theorem 4 implies the existence of a linear operator

One obtains

This concludes the proof.

In the next theorems, the notion of a resolvent appears naturally during the proofs.

Theorem 15. Let be as previously mentioned. Let , , , . Assume that

Then, there exists a linear positive operator , such that

Proof. The following implications hold

Note that in the preceding relations the Cauchy inequalities for the function were used. Application of Theorem 3 leads to the existence of a linear positive operator from into , satisfying the moment conditions, such that on . Now, the conclusion follows easily, by using the monotony of the norm on :

The proof is complete.

Similar results are valid for the multidimensional moment problem. Let be the space of analytic functions in the polydisc , . Whence,

The order relation on is defined by the positive cone

Consider an arbitrary real or complex Hilbert space commuting selfadjoint operators . We introduce the usual spaces:

Theorem 16 (see [8, Theorem 3.1]). With the previous notations, assume that

and let . Consider the following statements:

(a)there exists an -linear positive continuous operator such that

(b)one has

(c)one has

Then, (b)(a)(c).

Proof. (b)(a). Let be a finite subset and let be such that

Then, the order relation on and Cauchy’s inequalities yield

Using the hypothesis (b) and the previous relations, one obtains

Application of Theorem 3 leads to the existence of an -linear positive operator applying into such that

Using the monotony of the norm on , we deduce

Thus, the implication (b)(a) is proved.

(a)(c) The hypothesis on positivity of the solution and of functions yield

The proof is complete.

Applying Theorem 16 to the real Hilbert space , , , being the multiplication operator with the functions , , one obtains the following.

Corollary 17. Let , , .

Consider the following statements:

(a)there exists such that

(b)one has

(c)one has

Then,(b)(a)(c).

For some related results see [8].

Our next goal is to consider some Markov moment problems in terms of nonnegative sequences with respect to an interval. We recall this notion, that is well known.

Definition 18. A sequence in an ordered vector space is said to be nonnegative with respect to the interval if for any , we have

It is clear that for compact intervals and for , this condition is necessary and sufficient for the existence of a unique positive solution of the moment problem associated to the moments , . The continuity with respect to the supnorm is also obvious. But no information concerning dominating norm for the solution holds. Such information would be useful for integral representation of the extension of the solution to the space (see the next results).

Theorem 19. Let . Consider the following statements:

(a)there exists a unique such that

(b)the sequence

Then, (b)(a).

Proof. We have

Application of Theorem 4, (b)(a), leads to the existence of a linear functional on such that

Using the representation of a linear positive functional on , there exists

From the last equality written for , being a Borel subset, we infer (via measure theory) that

The proof is complete.

Lemma 20. Let be a positive function in and a continuous convex function on . Then, one has

Corollary 21. Let be a sequence of real numbers such that the sequence

is nonnegative with respect to and . Then,

Proof. Applying Theorem 19 and Lemma 20 to , , , one obtains

The proof is finished.

4. Extreme Points and the Markov Moment Problem

The aim of this section is to apply some ideas from [20, 26, 27] in solving truncated moment problems and their connection to the full moment problem. The results of this section are new. The following theorem concerns the finite Markov moment problem considered in [20], also using some results from [27]. Let us denote that

Theorem 22. For a given family of numbers , consider the following statements: (a), (b), and (c):

(a)there exists such that

(b)for any family of scalars , one has:

(c)there exists a Borel subset such that

Then,(b)(a)(c)

Proof. Let the point be accomplished and assume that

Then the integration on yields

Application of Theorem 4 to defined previously, leads to the existence of a linear positive form on such that

If characteristic functions of Borel subsets stand for , the conclusion (a) follows by measure theory [28]. The implication (a)(c) is a consequence of equality (15.14) in [27] (see also [27, Exercise 2.57]). The set of values for the control function , namely, , is replaced by , the set of values for , that stands for the control function . The extreme points of the positive part of the unit ball of are the characteristic functions of measurable sets. The converse is obvious.

Corollary 23. Under the equivalent conditions (a), (c) of Theorem 22, there are sequences

such that the following relations hold:

Proof. One uses the fact that any Borel subset is a joint of a set and a set of measure zero [28].

Remark 24. In order to approximate the numbers , one can make use of Fourier approximate expansion of with respect to the orthonormal sequence attached to the functions via Gram-Schmidt algorithm, also using the values of the moments . Thus, one obtains a smooth approximation of , and the intervals of ends are connected components of the open sets of the form

Remark 25. A similar result to that of Theorem 22 in several dimensions holds, with the same proof. We state it for the two-dimensional case.

Theorem 26. Let be a given family of real numbers and consider the functions

Consider the following assertions: (a), (b), (c). Then (b)(a)(c)

(a)there exists a Borel function a.e. such that

(b)for any family of scalars , one has

(c)there exists a Borel subset such that

Corollary 27. If one of the conditions (a), (c) from Theorem 26 is accomplished, there are sequences

such that

Proof. The Borel subset is the joint of a set and a null set. For an open subset , we consider its decomposition into cells [28], used in the construction of Lebesgue measure. If we determine smooth approximations of by means of Gram-Schmidt algorithm for the functions and the given moments via Fourier expansion, then the sequences from the present corollary can be determined by means of cell decomposition of the open subsets

Of course, this way one obtains approximations of these numbers.

The previous statements solve the truncated moment problems and sketch an algorithm for determining the numbers . In the last twenty years the truncated moment problem was studied intensively. In [26], the author proves that solving the truncated (not necessarily complex) moment problem for any natural yields to the solution of the full moment problem. The idea is of passing through the limit, which is allowed by a weakly compact argument. The following theorem proposes a similar construction, by using Krein-Milman theorem. We state it firstly in the one-dimensional case, although the several dimensional case one proves similarly.

Theorem 28. With the notations from Theorem 22, let be a sequence of real numbers. Consider the following assertions:

(a)there exists a Borel function such that

(b)for any natural number , and any , there exists nonnegative scalars , and sequences:

such as

(c)for any , there exists a Borel subset Bn such that

(d)for any natural and any , the following relation holds true:

Then (d)(a)(b)(c).

Proof. (a)(b). One applies Krein-Milman Theorem for the weakly compact subset formed by intersecting the unit ball of with the positive cone of the same space. Then, we must have

which are suitably chosen open subsets, where the limit is in the weak topology on , with respect to the dual pair ;  is the Lebesgue measure. This leads to

Since each open subset has an at most countable decomposition,

the conclusion follows. For the converse implication, observe that each step function,

is an element of the positive part of the unit ball in , and this subset is weakly compact. Let be the -limit of a subsequence of the sequence . Then, we have

by Lebesgue dominated convergence theorem. Hence, (b)(a) is proved. The implication (a)(c) follows from (a)(c) of Theorem 22, since a solution of the full moment problem is a solution of all truncated moment problems. The proof of (c)(a) is similar to that of (b)(a). It remains to prove that (d)(a). This is a consequence of the implication (b)(a) of Theorem 4. If is a finite subset and , then the following implications hold true:

Application of Theorem 4 leads to the existence of a linear functional on , verifying

Now, the conclusion follows by measure theory [28].

Remark 29. For the full moment problem, the following algorithm holds in determining .

Step  1. Approximating the function .

Let be a Hilbert base constructed by the aid of Gram-Schmidt procedure, applied to the system of linearly independent functions

Then, for each fixed , one has

where the coefficients are known from the Gram-Schmidt procedure. Hence, we can determine each Fourier coefficient of that is, we can approximate in -norm by a sequence of polynomial functions , . Then, there exists a subsequence [28] pointwise almost everywhere in .

Step  2. For each , the subsets,

are open subsets. Their connected components have end points approximations of the unknowns , . From [26], using a weakly compact standard argument, we can obtain as the limit of a subsequence of , where are as at point (c) of Theorem 28. Considering a suitable open set , from (a) one obtains:

Note that all the results of this section can be adapted to the multidimensional moment problem, with similar proofs.

5. Distanced Convex Subsets: Extension of Linear Operators and the Moment Problem

The case when the subset from Theorem 1 is distanced with respect to the subspace , while the restriction involving is missing, has a nice geometric meaning and leads to interesting results concerning the extension of linear functionals and operators (see [5]).

If is a convex neighborhood of the origin in a locally convex space, we denote by the gauge attached to .

Theorem 30. Let be a locally convex space, an order-complete vector lattice with strong order unit , and a vector subspace. Let be a convex subset with the following qualities:

(i)there exists a neighborhood of the origin such that ( and are distanced);(ii) is bounded.

Then, for any equicontinuous family of linear operators and for any , there exists an equicontinuous family such that

Moreover, if is a neighborhood of the origin such that

then the following relations hold:

The last relation of Theorem 30 gives a relationship between an upper bound and the lower bound of . The next result is a consequence of the preceding one, in terms of the moment problem. The topology on is the norm topology for which the unit closed ball is the interval .

Theorem 31. Let

Let be an order-complete vector lattice with strong order unit and a sequence in such that is positive with respect to . Then, for any , there exists such that

Moreover, if and is endowed with a linear topology such that its positive cone is closed and normal, then is continuous and positive.

Proof. Let

For all , we have

so that holds. This leads to , so we can take in Theorem 30  . From the hypothesis on , , we infer that

where is the constant from Theorem 30. We define that

Let . Using the hypothesis on the sequence which is positive with respect to , this yields

The conclusion is that verifies the conditions in the statement of Theorem 30. Application of the latter theorem leads to the existence of an extension of such that

Whence, the first part of the theorem is proved. It remains to prove the continuity and positivity of the extension , under the hypothesis mentioned in the statement. Replacing by , it is easy to see that

From these relations, also using that the positive cone is normal, the continuity of follows. In order to prove that is positive, it is sufficient to show that for all positive polynomials . Hypothesis yields

The proof is complete.

A consequence of Theorem 31 is the following one.

Corollary 32. Let be a Hilbert space, a positive selfadjoint operator, and the commutative algebra of selfadjoint operators defined in Section 2. Let be such that the spectrum and . Then, there exists a nondecreasing mapping such that

In particular, for such a mapping , the following relation holds:

Proof. We apply Theorem 31 to , , , . The sequence is positive with respect to because of the positivity of the spectral measure attached to . Due to the hypothesis of Theorem 31 being accomplished, there exists a positive continuous solution satisfying the relations in the statement of the latter theorem. This solution applies norm-bounded subsets into order-bounded subsets of . Application of [14, page 272], yields the existence of a “representing” nondecreasing mapping , such that the vector measure represents . Next, we prove the last relation of the statement:

Further computation yields

The proof is complete.

Application of Theorem 31 to positive sequences of real numbers with respect to , the functions , being as in Corollary 32, yields the following consequence, which we state without proof.

Corollary 33 (see [5, Corollary 3.3]). Let be a sequence of real numbers, which is positive with respect to , being arbitrary. Then, the following inequalities hold:

In particular, one has: .

Next, we consider the analogue of Theorem 31, for a space of analytic functions. Let and be the space of all functions which can be represented as an absolutely convergent series,

being continuous in the closed disc . Let , , being a measurable space, and for all .

Theorem 34. Let be a sequence such that

Assume that and let be a sequence in such that

Then, for any , there exists such that

Proof. If , we have

Let , , . Using the Cauchy inequalities, we derive

Define , , , . Using also the hypothesis on , one obtains

Whence, . Since

one takes in Theorem 30  , . Now, the conclusion follows via Theorem 30, for which all requirements of the hypothesis are accomplished.

6. Geometric Aspects Related to the Moment Problem

In the first part of this section, we give sufficient conditions for the existence of a solution, in a very general setting. The method works in an arbitrary measurable space, which may not involve polynomials. In the particular case of subsets of , the form of positive polynomials is not necessarily used.

Theorem 35. Let be a measurable space, where is a positive -finite measure on . Let , , . Consider the following statements:

(a)there exists such that

(b)for any finite subset and any , one has

(c)for any finite subset and any , one has

Then, (b)(a)(c).

Proof. (b)(a). The following implications hold:

On the other hand, the condition (b) can be rewritten as

From the preceding relation, we conclude

Whence, the conditions from the statement of Theorem 4 are accomplished, so that there exists a linear form on such that

Following the representation of continuous linear functionals on spaces, there exist

We also infer that

Thus, the proof of the implication (b)(a) is finished. The implication (a)(c) is almost obvious:

Taking the squares of the two members, the inequality is preserved. The proof is complete.

Corollary 36. Let

Consider the following statements:

(a)there exists a Lebesgue measurable function on such that

(b)for any finite subset and any , one has

Then, (b)(a).

Note that is a closed unbounded and non-semi-algebraic subset.

Details can be found in [8]. In [8, Theorem 2.5], the construction of the solution of a moment problem on the ellipse is given.

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