New Results on Markov Moment Problem

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 [0,∞), if the moments of arbitrary orders 𝑘(𝑘=0,1,2,…)   are given. Precisely, in the Stieltjes moment problem, a sequence of real numbers (𝑠_𝑘)_𝑘≥0is given, and one looks for a nondecreasing real function 𝜎(𝑡) (𝑡≥0) that verifies the moment conditions: ^∞₀𝑡^𝑘𝑑𝜎=𝑠_𝑘(𝑘=0,1,2,…). (1) 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 𝑅^𝑛, 𝑛≥2, 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 𝑓is a linear positive operator on a subspace 𝑆 of the ordered vector space 𝑋, the target space being an order-complete vector lattice 𝑌,   and for each 𝑥∈𝑋, there is 𝑠∈𝑆, 𝑥≤𝑠, then 𝑓 has a linear positive extension 𝐹∶𝑋→𝑌. 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 𝐴 with some properties with respect to the subspace 𝑆 (e.g., 𝑆∩ ri(𝐴)≠Φ), where ri(𝐴) is the relative interior of 𝐴. 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, 𝑋 will be a real vector space, 𝑌 an order-complete vector lattice, 𝐴,𝐵⊂𝑋 a convex subsets, 𝑞∶𝐴→𝑌 a concave operator, 𝑝∶𝐵→𝑌 a convex operator, 𝑆⊂𝑋 a vector subspace, 𝑓∶𝑆→𝑌 a linear operator.

Theorem 1. Assume that 𝑓∣_𝑆∩𝐴≥𝑞∣_𝑆∩𝐴, 𝑓∣_𝑆∩𝐵≤𝑝∣_𝑆∩𝐵.
The following assertions are equivalent:(a)there is a linear extension F∶X→Y of the operator f such that 𝐹∣_𝐴≥𝑞,𝐹∣_𝐵≤𝑝; (2)(b)there are p₁∶A→Y convex and q₁∶B→Y concave operators such that for all 𝜌,𝑡,𝜆^′,𝑎₁,𝑎^′,𝑏₁,𝑏^′,𝑣∈[0,1]²×(0,∞)×𝐴²×𝐵²×𝑆, (3)
One has:(1−𝑡)𝑎₁−𝑡𝑏₁=𝑣+𝜆^′1−𝜌𝑎^′−𝜌𝑏^′⇒(1−𝑡)𝑝₁𝑎₁−𝑡𝑞₁𝑏₁≥𝑓(𝑣)+𝜆^′1−𝜌𝑞𝑎^′−𝜌𝑝𝑏^′. (4)

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

Theorem 2. Let 𝑋 be a preordered vector space with its positive cone 𝑋₊,𝑝∶𝑋→𝑌 a convex operator, 𝑆⊂𝑋 a vector subspace, and 𝑓∶𝑆→𝑌 a linear positive operator. The following assertions are equivalent:(a)there is a linear positive extension F∶X→Y of f such that 𝐹(𝑥)≤𝑝(𝑥)∀𝑥∈𝑋; (5)(b)f(x^′)≤p(x) for all (x^′,x)∈S×X such that x^′≤x.

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

Theorem 3. Let 𝑋, 𝑌, 𝑝∶𝑋→𝑌 be as in Theorem 2, {𝑥_𝑗}_𝑗∈𝐽⊂𝑋, {𝑦_𝑗}_𝑗∈𝐽⊂𝑌 given families. The following statements are equivalent: (a)there is a linear positive operator 𝐹∶𝑋→𝑌 such that 𝐹𝑥_𝑗=𝑦_𝑗∀𝑗∈𝐽,𝐹(𝑥)≤𝑝(𝑥)∀𝑥∈𝑋; (6)(b)for any finite subset 𝐽₀⊂𝐽 and any {𝜆_𝑗}_𝑗∈𝐽0⊂𝑅, one has _𝑗∈𝐽0𝜆_𝑗𝑥_𝑗≤𝑥⇒_𝑗∈𝐽0𝜆_𝑗𝑦_𝑗≤𝑝(𝑥). (7)
A clearer sandwich-moment problem variant is the following one.

Theorem 4. Let 𝑋,  𝑌,  {𝑥_𝑗}_𝑗∈𝐽,{𝑦_𝑗}_𝑗∈𝐽 be as in Theorem 3 and 𝐹₁, 𝐹₂∈𝐿(𝑋,𝑌) two linear operators. The following statements are equivalent:(a)there is a linear operator F∈L(X,Y) such that 𝐹₁(𝑥)≤𝐹(𝑥)≤𝐹₂(𝑥)∀𝑥∈𝑋₊,𝐹𝑥_𝑗=𝑦_𝑗∀𝑗∈𝐽; (8)(b)for any finite subset 𝐽₀⊂𝐽 and any {𝜆_j}_j∈J0⊂R, one has: _𝑗∈𝐽0𝜆_𝑗𝑥_𝑗=𝜑₂−𝜑₁,𝜑₁,𝜑₂∈𝑋₊⇒_𝑗∈𝐽0𝜆_𝑗𝑦_𝑗≤𝐹₂𝜑₂−𝐹₁𝜑₁. (9)

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 𝑥∈𝐶₀([0,∞)×[0,∞)) is a nonnegative continuous function with compact support, then there exists a sequence (𝑝_𝑚)_𝑚 of positive polynomials on [0,∞)×[0,∞), such that 𝑝_𝑚(𝑡)>𝑥(𝑡)∀𝑡≥0,∀𝑚∈𝑍₊,𝑝_𝑚→𝑥 (10) uniformly on compact subsets of [0,∞)×[0,∞).

The idea of the proof is to add the ∞ point and to apply the Stone-Weierstrass theorem to the subalgebra generated by the functions exp(−𝑚𝑡₁−𝑛𝑡₂),  𝑚,𝑛∈𝑍₊. Then, one uses for each such exp-function suitable majorizing or minorizing partial polynomial sums, as well as the relation (𝑠)−𝑠exp1++𝑠1!²𝑠2!+⋯+^𝑚=(𝑠)𝑚!exp𝑚!^𝑠₀exp(−𝑡)⋅𝑡^𝑚𝑑𝑡,𝑚∈𝑍₊. (11) Note that [10, Lemma 1.4] asserts the density of positive polynomials in (𝐿¹_𝜈(𝐴))₊, for any closed subset 𝐴 of a finite-dimensional space, 𝜈, being a positive regular Borel M-determinate measure. Here is the exact formulation of this result.

Lemma 6 (see [10, Lemma 1.4]). Let 𝐴⊂𝑅^𝑛 be an arbitrary closed subset and 𝜈 an M-determinate positive regular Borel measure on 𝐴, with finite moments of all natural orders. Then, for any 𝑥∈(𝐶₀(𝐴))₊ such that 𝑥≤𝑝∈𝑃 for some polynomial 𝑝, there exists a sequence (𝑝_𝑚)_𝑚,𝑝_𝑚∈𝑃₊,𝑝_𝑚≥𝑥, 𝑝_𝑚→𝑥 in 𝐿¹_𝜈(𝐴). In particular, one has: lim_𝐴𝑝_𝑚(𝑡)𝑑𝜈=_𝐴𝑥(𝑡)𝑑𝜈, (12)𝑃₊ is dense in (𝐿¹_𝜈(𝐴))₊, and 𝑃 is dense in 𝐿¹_𝜈(𝐴),

Recall that 𝑃 is the subspace of all polynomial functions on 𝐴.

Lemma 7. For any simple function of the form 𝑠=^𝑁_𝑗=1𝛼_𝑗𝜒_𝐼𝑗,𝛼_𝑗≥0𝑗=1,…,𝑁,𝐼_𝑗⊂[0,∞)𝑗=1,…,𝑁, (13) being intervals, there exists a sequence of polynomial 𝑝_𝑚𝑚,𝑝_𝑚→𝑠in𝐿¹_𝜈([0,∞)),𝑝_𝑚≥𝑠∀𝑚, (14) where 𝜈 is an M-determinate positive regular Borel measure with finite moments of all natural orders. If 𝑠 is extended to an even function in 𝑅, then the polynomials 𝑝_𝑚,𝑚∈𝑍₊ are restrictions to the positive semiaxes of even polynomials in 𝑅.

Note that in the preceding Lemmas 5–7, the sequence (𝑝_𝑚)_𝑚 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 𝑝₂(𝑡)=𝑝²_2,1𝑡²+𝑡²𝑝²_2,2𝑡²,𝑡∈𝑅,𝑝_2,𝑗∈𝑅[𝑡],𝑗=1,2. (15) We start by proving some results on the one-dimensional moment problem.

Theorem 9 (see [10, Theorem 2.1]). Let 𝜈 be a positive determinate Borel regular measure on 𝑅 with finite moments 𝑚_2𝑘=∫_𝑅𝑡^2𝑘𝑑𝜈(𝑡²) of all orders, 𝑘∈𝑍₊ such that ∫_𝑅𝑥₁(𝑡)𝑑𝜈=0 for any odd function 𝑥₁∈𝐿¹_𝜈(𝑅). Let (𝑦_2𝑘)_𝑘∈𝑍+ be a given sequence of real numbers.
The following statements are equivalent:(a)there exists a Borel function ℎ such that _𝑅𝑡^2𝑘ℎ𝑡²𝑑𝜈𝑡²=𝑦_2𝑘,_𝑅𝑡^2𝑘+1ℎ𝑡²𝑑𝜈=0,∀𝑘∈𝑍₊,0≤ℎ𝑡²≤1𝑑𝜈-a.e.in𝑅; (16)(b)for any finite subset J₀⊂Z₊ and any {𝜆_j;j∈J₀}⊂R, one has 0≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝑦_2𝑖+2𝑗≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝑅𝑡^2𝑖+2𝑗𝑑𝜈,0≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝑦_{2𝑖+2𝑗+2}≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝑅𝑡^{2𝑖+2𝑗+2}𝑑𝜈. (17)

Proof. The hypothesis (a) implies that (b) is obvious because of the qualities of ℎ.
(b)⇒(a). Since 𝜈 vanishes on the subspace of odd functions, the same property is valid for ℎ(𝑡²)d𝜈. The problem can be reduced to a corresponding one on the positive semiaxis 𝑢=𝑡²≥0. Using the form of the positive polynomials in 𝑢∈𝑅₊ [1], the hypothesis (b) yields𝑓_𝑗∈𝐽0𝜆_𝑗𝑥_𝑗∶=_𝑗∈𝐽0𝜆_𝑗𝑦_2𝑗2,𝑥_𝑗(𝑢)=𝑢^𝑗,𝑗∈𝑍₊⇒0≤𝑓𝑝≤^∞₀𝑝(𝑢)𝑑𝜈∀𝑝∈𝑃₊. (18) Since 𝑃 is a majorizing subspace in 𝑋∶=𝑥∈𝐿¹_𝜈([0,∞));∃𝑝∈𝑃,|𝑥(𝑢)|≤𝑝(𝑢),∀𝑢≥0, (19) there exists a positive linear extension 𝐹∈𝑋^∗ of 𝑓. By Haviland-Riesz representation theorem, there exists a representing positive regular Borel measure 𝜇 on [0,∞) such that 𝜇𝑝=𝐹𝑝=𝑓𝑝≤𝜈𝑝,𝑝∈𝑃₊. (20) Applying Haviland-Riesz theorem once more, we infer the existence of a positive regular Borel measure 𝜆 on [0,∞), such that 𝜆𝑝=𝜈−𝜇𝑝⇒𝜈𝑝=𝜆𝑝+𝜇𝑝,𝑝∈𝑃. (21) Since 𝜈 is M-determinate, application of Lemma 7 and the last relation lead to 𝜈(𝐵)=𝜆(𝐵)+𝜇(𝐵),∀𝐵⊂[0,∞), (22) Borel subset. In particular, 𝜇 is absolutely continuous with respect to 𝜈, hence there exists ℎ₁∈𝐿¹_𝜈([0,∞)) such that 𝑑𝜇=ℎ₁𝑑𝜈 on simple functions. From [28, Theorem 1.40], we infer that 0≤ℎ₁≤1𝜈-a.e. Since 𝜈 was a positive regular Borel measure, we can find a Borel function ℎ such that ℎ=ℎ₁𝜈-a.e. The previous relations lead to ^∞₀𝑢^𝑗ℎ(𝑢)𝑑𝜈=^∞₀𝑢^𝑗𝑑𝜇=𝐹𝑥_𝑗=𝑦_2𝑗2⇒^∞_−∞𝑡^2𝑗ℎ𝑡²𝑑𝜈𝑡²𝑦=2^2𝑗2=𝑦_2𝑗 (23) This concludes the proof.

Corollary 10. Let 𝛼>0 and {𝑦_2𝑘}_𝑘∈𝑍+⊂𝑅. The following statements are equivalent:(a)there exists a Borel function h, such that _𝑅𝑡^2𝑘ℎ𝑡²𝑑𝑡=𝑦_2𝑘,_𝑅𝑡^2𝑘+1ℎ𝑡²𝑑𝑡=0,∀𝑘∈𝑍₊,0≤ℎ𝑡²≤1√−𝑡2𝜋𝛼exp²2𝛼²a.e.in𝑅; (24)(b)for any finite subset J₀⊂Z₊ and any {𝜆_j;j∈J₀}⊂R, one has0≤_{𝑖,𝑗∈𝐽}𝜆_𝑖𝜆_𝑗𝑦_2𝑖+2𝑗≤_{𝑖,𝑗∈𝐽}𝜆_𝑖𝜆_𝑗𝛼^2(𝑖+𝑗)2𝑖+𝑗−1!!,0≤_{𝑖,𝑗∈𝐽}𝜆_𝑖𝜆_𝑗𝑦_{2𝑖+2𝑗+2}≤_{𝑖,𝑗∈𝐽}𝜆_𝑖𝜆_𝑗𝛼^{2(𝑖+𝑗+1)}2𝑖+𝑗+1−1!!. (25)

Proof. One applies Theorem 9 to 𝑑𝜈=(1/(√2𝜋𝛼))exp(−𝑡²/2𝛼²)𝑑𝑡, as a measure on 𝑅.
Due to Carleman criterion, 𝑑𝜈 is M-determinate. A straightforward computation shows that _𝑅𝑡^2𝑘𝑑𝜈=𝛼^2𝑘(2𝑘−1)!!,𝑘≥1,𝑘∈𝑍₊. (26) The conclusion follows.

Using a quite different proof with respect to that of Theorem 9, we proved the following “abstract operatorial version.” Let 𝜈 be a positive measure as in Theorem 9, vanishing on the subspace of odd functions from 𝐿¹_𝜈(𝑅). Let 𝑌 be a Banach lattice, which is order complete. Let 𝑥_𝑗(𝑡)=𝑡^𝑗, 𝑡∈𝑅, 𝑡²=𝑢∈[0,∞), 𝑥_𝑗(𝑢)=𝑢^𝑗. Then, the following theorem holds.

Theorem 11 (see [10, Theorem 2.1]). Let 𝐹₂∶𝐿¹_𝜈(𝑅)→𝑌 be a linear continuous positive operator vanishing on the subspace of odd functions and {𝑦_2𝑘}_𝑘⊂𝑌.
The following statements are equivalent:(a)there exists a linear operator F∶L¹_𝜈(R)→Y such that 𝐹𝑥_2𝑘=𝑦_2𝑘,𝐹𝑥_2𝑘+1=0,𝑘∈𝑍₊;0≤𝐹(𝑥)≤𝐹₂(𝑥)∀𝑥∈𝐿¹_𝜈(𝑅)₊⇒‖𝐹‖≤𝐹₂; (27)(b)for any finite subset J₀⊂Z₊ and any {𝜆_j;j∈J₀}⊂R, one has 0≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝑦_2𝑖+2𝑗≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝐹₂𝑥_2𝑖+2𝑗,0≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝑦_{2𝑖+2𝑗+2}≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝐹₂𝑥_{2𝑖+2𝑗+2}. (28)

In the sequel, 𝑌 will be the order-complete vector lattice, which is also a commutative Banach algebra of selfadjoint operators [6, 14]. Let 𝐻 be a Hilbert space 𝐴 a self-adjoint operator from 𝐻 into 𝐻, 𝑌₁=𝑌₁(𝐴)={𝑇∈Α(𝐻);𝐴𝑇=𝑇𝐴},𝑌=𝑇∈𝑌₁;𝑇𝑈=𝑈𝑇,∀𝑈∈𝑌₁. (29) Let 𝑋 be the space of all continuous functions 𝑥 applying 𝑅 into 𝑅, such that |𝑥| is majorized by a polynomial on 𝑅. Denote by 𝑋₂ the subspace of even functions, and 𝑆(𝐴) will be the spectrum of 𝐴.

Theorem 12. Let {𝐵_2𝑘}_𝑘∈𝑍+⊂𝑌=𝑌(𝐴). The following statements are equivalent:(a)there exists 𝐹∈𝐿(𝑋,𝑌) such that𝐹𝑥_2𝑘=𝐵_2𝑘,𝐹𝑥_2𝑘+1=0∀𝑘∈𝑍₊,0≤𝐹𝜑₂⋅𝜒_𝑆(𝐴)≤_𝑆(𝐴)𝜑₂(𝑡)𝑑𝐸_𝐴,𝜑∈𝑋_2,+; (30)(b)for any finite subset 𝐽₀⊂𝑍₊ and any {𝜆_𝑗;𝑗∈𝐽₀}⊂𝑅, one has0≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝐵_2𝑖+2𝑗≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝐴^2𝑖+2𝑗,0≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝐵_{2𝑖+2𝑗+2}≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝐴^{2𝑖+2𝑗+2}. (31)

Proof. (b)⇒(a). Let 𝑓∶𝑃∩𝑋₂→𝑌,𝑓_𝑗∈𝐽0𝜆_𝑗𝑥_2𝑗=_𝑗∈𝐽0𝜆_𝑗𝐵_2𝑗. (32) Then, the assumptions of (b) become 0≤𝑓𝑝₂≤_𝑆(𝐴)𝑝₂(𝑡)𝑑𝐸_𝐴,∀𝑝₂∈𝑃_2,+. (33) Since 𝑃₂ is a majorizing subspace in 𝑋₂, 𝑓 has a positive linear extension 𝐹₂∈𝐿₊𝑋₂,𝑌. (34)
Define 𝐹∶𝑋→𝑌,𝐹(𝑥)=𝐹𝑥₁+𝑥₂=𝐹₂𝑥₂,𝑥₂(𝑡)=12(𝑥(𝑡)+𝑥(−𝑡)),𝑡∈𝑅. (35) Then, 𝐹 is a linear positive extension of 𝑓 to the space 𝑋. The first two properties of the solution 𝐹 hold. Using Urysohn’s lemma, for any 𝜑₂∈(𝑋₂)₊, there is a sequence 𝜑_2,𝑙𝑙⊂𝐶_𝑐(𝑅)₊∩𝑋₂,𝜑_2,𝑙↓𝜑₂⋅𝜒_𝑆(𝐴). (36) Applications of the one-dimensional variant of Lemma 5 lead to the existence of a sequence of polynomial functions (𝑝_𝑚)_𝑚 such that 𝑝_𝑚∣_𝑆(𝐴)→𝜑₂𝜒_𝑆(𝐴)uniformlyon𝑆(𝐴). (37) These arguments and the positivity of 𝐹 yield 0≤𝐹𝜑₂⋅𝜒_𝑆(𝐴)≤inf𝐹𝑝_𝑚=inf𝑓𝑝_𝑚≤inf_𝑆(𝐴)𝑝_𝑚(𝑡)𝑑𝐸_𝐴=_𝑆(𝐴)𝜑₂(𝑡)𝑑𝐸_𝐴. (38) 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 𝑋=𝑥∶𝑅²𝑥𝑡→𝑅;𝑥continuous,∃𝑝∈𝑃s.t.₁,𝑡₂≤𝑝𝑡₁,𝑡₂,∀𝑡₁,𝑡₂∈𝑅²,𝑋₂=𝑥∈𝑋;𝑥𝑡₁,𝑡₂=𝑥±𝑡₁,±𝑡₂,𝑡₁,𝑡₂∈𝑅². (39) If 𝐻 is a Hilbert space and 𝐴₁,𝐴₂∈Α(𝐻) are two bounded positive commuting selfadjoint operators, one defines 𝑌₁=𝑌₁𝐴₁,𝐴₂=𝑇∈Α(𝐻);𝑇𝐴_𝑗=𝐴_𝑗𝑇,𝑗=1,2,𝑌=𝑈∈𝑌₁;𝑇𝑈=𝑈𝑇,∀𝑇∈𝑌₁. (40) Then, 𝑌 endowed with the usual order relation on selfadjoint operators is an order-complete vector lattice and a commutative Banach algebra.

Theorem 13. Let {𝐵_{(2𝑚,2𝑛)}}_{(𝑚,𝑛)∈𝑍2+}⊂𝑌(𝐴₁,𝐴₂) be a given sequence. The following statements are equivalent:(a)there exists 𝐹∈𝐿(𝑋,𝑌) such that𝐹𝑥_{(2𝑚,2𝑛)}=𝐵_{(2𝑚,2𝑛)},(𝑚,𝑛)∈𝑍²₊,𝐹𝑥_(𝑘,𝑙)=0,∀(𝑘,𝑙)∈𝑍₊²⧵2𝑍₊²,0≤𝐹𝑥₂≤_{𝑆(𝐴1)×𝑆(𝐴2)}𝑥₂𝑡₁,𝑡₂𝑑𝐸_𝐴1𝑑𝐸_𝐴2∀𝑥₂∈𝑋₂₊,𝐹𝜑≤_{𝑆(𝐴1)×𝑆(𝐴2)}𝜑𝑡₁,𝑡₂𝑑𝐸_𝐴1𝑑𝐸_𝐴2∀𝜑∈𝑋₂; (41)(b)for all 𝐽₁,𝐽₂⊂𝑍₊ finite subsets, for all {𝑎_𝑗;𝑗∈𝐽₁}, and for all {𝑏_𝑚;𝑚∈𝐽₂}⊂𝑅, one has0≤_{𝑖,𝑗∈𝐽1,𝑚,𝑛∈𝐽2}𝑎_𝑖𝑎_𝑗𝑏_𝑚𝑏_𝑛𝐵_{(2𝑖+2𝑗,2𝑚+2𝑛)}≤_{𝑖,𝑗∈𝐽1,𝑚,𝑛∈𝐽2}𝑎_𝑖𝑎_𝑗𝑏_𝑚𝑏_𝑛𝐴₁^2𝑖+2𝑗𝐴₂^2𝑚+2𝑛0≤_{𝑖,𝑗∈𝐽1,𝑚,𝑛∈𝐽2}𝑎_𝑖𝑎_𝑗𝑏_𝑚𝑏_𝑛𝐵_{(2𝑖+2𝑗+2,2𝑚+2𝑛)}≤_{𝑖,𝑗∈𝐽1,𝑚,𝑛∈𝐽2}𝑎_𝑖𝑎_𝑗𝑏_𝑚𝑏_𝑛𝐴₁^{2𝑖+2𝑗+2}𝐴₂^2𝑚+2𝑛0≤_{𝑖,𝑗∈𝐽1,𝑚,𝑛∈𝐽2}𝑎_𝑖𝑎_𝑗𝑏_𝑚𝑏_𝑛𝐵_{(2𝑖+2𝑗,2𝑚+2𝑛+2)}≤_{𝑖,𝑗∈𝐽1,𝑚,𝑛∈𝐽2}𝑎_𝑖𝑎_𝑗𝑏_𝑚𝑏_𝑛𝐴₁^2𝑖+2𝑗𝐴₂^{2𝑚+2𝑛+2}0≤_{𝑖,𝑗∈𝐽1,𝑚,𝑛∈𝐽2}𝑎_𝑖𝑎_𝑗𝑏_𝑚𝑏_𝑛𝐵_{(2𝑖+2𝑗+2,2𝑚+2𝑛+2)}≤_{𝑖,𝑗∈𝐽1,𝑚,𝑛∈𝐽2}𝑎_𝑖𝑎_𝑗𝑏_𝑚𝑏_𝑛𝐴₁^{2𝑖+2𝑗+2}𝐴₂^{2𝑚+2𝑛+2}. (42)

Proof. Since (a)⇒(b) is obvious, we have to prove the implication (b)⇒(a). By Stone-Weierstrass theorem, we know that the vector subspace 𝑆 generated by the functions 𝑓𝑡₁,𝑡₂=𝑓₁𝑡₁⋅𝑓₂𝑡₂=𝑓₁⊗𝑓₂𝑡₁,𝑡₂,𝑓_𝑗∈𝐶𝑆𝐴_𝑗,𝑗=1,2, (43) is a dense subalgebra of 𝐶(𝑆(𝐴₁)×𝑆(𝐴₂)). Hence, for any 𝑥∈(𝐶(𝑆(𝐴₁)×𝑆(𝐴₂)))₊, there exists a sequence (𝑓_𝑚)_𝑚 of elements from 𝑆, which converges uniformly to 𝑥 on 𝑆(𝐴₁)×𝑆(𝐴₂), such that 𝑓_𝑚>𝑥≥0 in 𝑆(𝐴₁)×𝑆(𝐴₂). For any function from 𝐶(𝑆(𝐴₁)×𝑆(𝐴₂)), we consider its extension to [0,∞)×[0,∞) which vanishes outside 𝑆(𝐴₁)×𝑆(𝐴₂), and then we extend again this function to 𝑅², such that 𝑥(𝑡₁,𝑡₂)=𝑥(±𝑡₁,±𝑡₂), for all (𝑡₁,𝑡₂)∈𝑅². If we write 𝑓_𝑚𝑡₁,𝑡₂=𝑓_𝑚,1𝑡₁𝑓_𝑚,2𝑡₂,𝑡₁,𝑡₂∈𝑅², (44) where 𝑓_𝑚,𝑗 are extended to 𝑅₊ such that to vanish outside 𝑆(𝐴_𝑗),𝑗=1,2, and to 𝑅 such that to be even functions, we infer that 𝑓_𝑚,1,𝑓_𝑚,2 are even bounded measurable nonnegative functions on 𝑅 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 𝑓_𝑚,𝑗 on 𝑆(𝐴_𝑗),𝑗=1,2, by even positive polynomials on 𝑅. Using the representation of these polynomials given in Remark 8, (b) is rewritten as 𝑓𝑎_𝑗𝑡₁^2𝑗𝑏_𝑚𝑡₂^2𝑚∶=𝑎_𝑗𝑏_𝑚𝐵_{(2𝑗,2𝑚)}⇒0≤𝑓𝑝₁𝑝₂≤_{𝑆(𝐴1)×𝑆(𝐴2)}𝑝₁𝑡₁⋅𝑝₂𝑡₂𝑑𝐸_𝐴1𝑑𝐸_𝐴2,𝑝_𝑗∈𝑅𝑡_𝑗+,𝑗=1,2. (45) In particular, 𝑓 is a linear positive operator on the subspace generated by the products of even polynomials in each variable, which is a majorizing subspace in 𝑋₂. This yields the existence of a positive extension 𝐹 of 𝑓 to the whole space 𝑋₂. The positivity of 𝐹 and the uniform convergence of (𝑝_𝑚)_𝑚 to 𝑥₂≥0 on 𝑆(𝐴₁)×𝑆(𝐴₂) lead to 𝐹𝑥₂=lim𝐹𝑝_𝑚=lim𝑓𝑝_𝑚≤lim_{𝑆(𝐴1)×𝑆(𝐴2)}𝑝_𝑚,1𝑡₁𝑝_𝑚,2𝑡₂𝑑𝐸_𝐴1𝑑𝐸_𝐴2=_{𝑆(𝐴1)×𝑆(𝐴2)}𝑥₂𝑡₁,𝑡₂𝑑𝐸_𝐴1𝑑𝐸_𝐴2. (46) 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, 𝑋 will be the space of all absolutely convergent power series:𝜑(𝑧)=^∞_𝑗=0𝛼_𝑗𝑧^𝑗,|𝑧|𝑋<𝑏,₊=𝜑∈𝑋;𝛼_𝑗∈𝑅₊∀𝑗∈𝑍₊,𝑋=𝑋₊−𝑋₊. (47) One assumes that the elements of the space 𝑋 are continuous in the closed disk {|𝑧|≤𝑏}. Let 𝐻 be a Hilbert space, 𝑈₀ a selfadjoint operator acting on 𝐻, and 𝑌=𝑌(𝑈₀) the order-complete vector lattice and commutative Banach algebra of selfadjoint operators defined in Section 2 (see also [6, 14]). We denote that 𝜑_𝑗(𝑧)=𝑧^𝑗, |𝑧|≤𝑏, 𝑗∈𝑍₊.

Theorem 14. Let 𝑋, 𝐻, 𝑈₀, 𝑌, 𝜑_𝑗 be as previously mentioned. Assume that 𝑏>1 and let 𝐴∈𝑌, ‖𝐴‖<𝑏. Let 𝜀>0 and {𝐵_𝑗}_𝑗∈𝑍+⊂𝑌. The following statements are equivalent:(a)there exists a linear operator 𝐹 applying 𝑋 into 𝑌 such that𝐹𝜑_𝑗=𝐵_𝑗∀𝑗∈𝑍₊,𝜑(𝐴)−𝜀⋅𝜑(𝐼)≤𝐹𝜑≤𝜑(𝐴)+𝜀⋅𝜑(𝐼)∀𝜑∈𝑋₊,𝐹𝜑≤𝜑+𝜀⋅𝜑(1),𝜑∈𝑋; (48)(b)one has𝐴^𝑗−𝜀⋅𝐼≤𝐵_𝑗≤𝐴^𝑗+𝜀⋅𝐼,𝑗∈𝑍₊. (49)

Proof. (a)⇒(b) is obvious, since from (a) we derive 𝐵_𝑗=𝐹𝜑_𝑗∈𝜑_𝑗(𝐴)−𝜀⋅𝜑_𝑗(𝐼),𝜑_𝑗(𝐴)+𝜀⋅𝜑_𝑗(𝐼=𝐴_𝑗)−𝜀⋅𝐼,𝐴^𝑗+𝜀⋅𝐼,𝑗∈𝑍₊. (50)
(b)⇒(a). One applies Theorem 4 to 𝑥_𝑗=𝜑_𝑗∈𝑋,𝑗∈𝑍₊. We have _𝑗∈𝐹𝜆_𝑗𝜑_𝑗=𝜓₂−𝜓₁=_𝑗∈𝑍+𝑎_𝑗𝜑_𝑗−_𝑗∈𝑍+𝑏_𝑗𝜑_𝑗,𝑎_𝑗,𝑏_𝑗𝑎≥0⇒−_𝑗+𝑏_𝑗≤−𝑏_𝑗≤𝜆_𝑗≤𝑎_𝑗≤𝑎_𝑗+𝑏_𝑗⇒𝜆_𝑗≤𝑎_𝑗+𝑏_𝑗,𝑗∈𝐹⇒𝜆_𝑗𝐵_𝑗≤𝜆_𝑗𝐴^𝑗+𝜀⋅𝜆_𝑗𝐼,∀𝑗∈𝐹⁺,𝜆_𝑗𝐵_𝑗≤𝜆_𝑗𝐴^𝑗−𝜀⋅𝜆_𝑗𝐼,∀𝑗∈𝐹⁻,𝐹⁺=𝑗∈𝐹;𝜆_𝑗≥0,𝐹⁻=𝑗∈𝐹;𝜆_𝑗<0. (51)
The preceding relations yield _𝑗∈𝐹𝜆_𝑗𝐵_𝑗≤_𝑗∈𝐹+𝜆_𝑗𝐴^𝑗+𝜀⋅_𝑗∈𝐹+𝜆_𝑗⋅𝐼+_{𝑗∈𝐹−}𝜆_𝑗𝐴^𝑗−𝜀⋅_{𝑗∈𝐹−}𝜆_𝑗⋅𝐼=_𝑗∈𝐹𝜆_𝑗𝐴^𝑗+𝜀⋅_𝑗∈𝐹𝜆_𝑗⋅𝐼=𝜓₂(𝐴)−𝜓₁(𝐴)+𝜀⋅_𝑗∈𝐹𝜆_𝑗⋅𝐼≤𝜓₂(𝐴)−𝜓₁(𝐴)+𝜀⋅_𝑗∈𝑍+𝜆_𝑗⋅𝐼≤𝜓₂(𝐴)−𝜓₁(𝐴)+𝜀⋅_𝑗∈𝑍+𝑎_𝑗+𝑏_𝑗⋅𝐼=𝜓₂(𝐴)−𝜓₁(𝐴)+𝜀⋅_𝑗∈𝑍+𝑎_𝑗+𝑏_𝑗⋅𝐼=𝜓₂(𝐴)+𝜀⋅_𝑗∈𝑍+𝑎_𝑗−𝜓⋅𝐼₁(𝐴)−𝜀⋅_𝑗∈𝑍+𝑏_𝑗⋅𝐼=𝐹₂𝜓₂−𝐹₁𝜓₁. (52) Theorem 4 implies the existence of a linear operator 𝐹∈𝐿(𝑋,𝑌),𝐹₁≤𝐹≤𝐹₂on𝑋₊,𝐹𝜑_𝑗=𝐵_𝑗∀𝑗∈𝑍₊. (53)
One obtains 𝐹𝜑≤𝐹𝜑₊+𝐹𝜑₋≤𝜑₊(𝐴)+𝜀⋅𝜑₊(𝐼)+𝜑₋(𝐴)+𝜀⋅𝜑₋(𝐼)=𝜑(𝐴𝜑(𝐼≤𝜑𝜑(1⇒𝐹𝜑≤𝜑𝜑(1)+𝜀)⋅𝐼+𝜀⋅)⋅𝐼+𝜀⋅),∀𝜑∈𝑋. (54) This concludes the proof.

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

Theorem 15. Let 𝑏>1,𝑋,𝑌,{𝜑_𝑗}_𝑗 be as previously mentioned. Let 𝐴∈𝑌, 𝑆(𝐴)⊂(0,𝑏), {𝐵_𝑗}_𝑗∈𝑍+⊂𝑌, 𝜀>0. Assume that0≤𝐵_𝑗≤𝐴^𝑗+𝜀⋅𝐼∀𝑗∈𝑍₊. (55)Then, there exists a linear positive operator 𝐹∈𝐿₊(𝑋,𝑌), such that𝐹𝜑_𝑗=𝐵_𝑗∀𝑗∈𝑍₊,𝐹𝜑≤𝜑⋅𝐼−𝑏⁻¹𝐴⁻¹𝑏+𝜀𝐼⇒‖𝐹‖≤𝑏𝑏−1‖𝐴‖𝑏𝑏−+𝜀.𝑏−1 (56)

Proof. The following implications hold _𝑗∈𝐹𝜆_𝑗𝜑_𝑗≤𝜑=_𝑗∈𝑍+𝑎_𝑗𝜑_𝑗⇒𝜆_𝑗≤𝑎_𝑗≤𝑎_𝑗≤𝜑_∞𝑏^𝑗⇒,𝑗∈𝐹_𝑗∈𝐹𝜆_𝑗𝐵_𝑗≤_𝑗∈𝐹+𝜆_𝑗𝐴^𝑗≤𝜑⋅+𝜀⋅𝐼_𝑗∈𝑍+𝑏^−𝑗𝐴^𝑗+𝜀⋅_𝑗∈𝑍+𝑏^−𝑗=𝜑⋅⋅𝐼𝐼−𝑏⁻¹𝐴⁻¹𝑏+𝜀⋅𝐼𝜑𝑏−1=𝑝,𝜑∈𝑋. (57) Note that in the preceding relations the Cauchy inequalities for the function 𝜑 were used. Application of Theorem 3 leads to the existence of a linear positive operator 𝐹 from 𝑋 into 𝑌, satisfying the moment conditions, such that 𝐹≤𝑝 on 𝑋. Now, the conclusion follows easily, by using the monotony of the norm on 𝑌: ‖𝐹‖≤𝐼−𝑏⁻¹𝐴⁻¹𝑏+𝜀⋅‖𝐴‖𝑏−1≤1+𝑏+‖𝐴‖²𝑏²𝑏+⋯+𝜀⋅=𝑏𝑏−1‖𝐴‖𝑏𝑏−+𝜀⋅.𝑏−1 (58) The proof is complete.

Similar results are valid for the multidimensional moment problem. Let 𝑋_𝜌 be the space of analytic functions 𝜑 in the polydisc 𝑇_𝜌=∏^𝑛_𝑘=1{|𝑧_𝑘|≤𝜌_𝑘}, 𝜌=(𝜌₁,…,𝜌_𝑛). Whence, 𝜑𝑧₁,…,𝑧_𝑛=_{𝑗∈𝑍𝑛+}𝑐_𝑗𝑧^𝑗=_{𝑗∈𝑍𝑛+}𝑐_𝑗𝑧^𝑗1₁⋯𝑧^𝑗𝑛_𝑛,𝑗=𝑗₁,…,𝑗_𝑛,𝑧_𝑗≤𝜌_𝑗𝜑,𝑗=1,…,𝑛,_𝑗𝑧₁,…,𝑧_𝑛=𝑧^𝑗1₁⋯𝑧^𝑗𝑛_𝑛. (59) The order relation on 𝑋 is defined by the positive cone 𝑋₊=𝜑∈𝑋;Rec_𝑗≥0,Im𝑐_𝑗≥0. (60) Consider an arbitrary real or complex Hilbert space 𝐻,𝐴₁,…,𝐴_𝑛∈𝐴(𝐻) commuting selfadjoint operators (𝐴_𝑖𝐴_𝑗=𝐴_𝑗𝐴_𝑖,𝑖,𝑗=1,…,𝑛). We introduce the usual spaces: 𝑌₁=𝑈∈𝐴(𝐻);𝑈𝐴_𝑗=𝐴_𝑗𝑈,𝑗=1,…,𝑛,𝑌=𝑈∈𝑌₁;𝑈𝑉=𝑉𝑈,∀𝑉∈𝑌₁,𝑌₊={𝑈∈𝑌;⟨𝑈(ℎ),ℎ⟩≥0,∀ℎ∈𝐻}. (61)

Theorem 16 (see [8, Theorem 3.1]). With the previous notations, assume that 𝜌_𝑗>𝐴_𝑗,𝑗=1,…,𝑛, (62) and let {𝐵_𝑗;𝑗∈𝑍^𝑛₊}⊂𝑌. Consider the following statements:(a)there exists an R-linear positive continuous operator F∶X_𝜌→Y such that 𝐹𝑥_𝑗=𝐵_𝑗,𝑗∈𝑍^𝑛₊,𝐹𝜑≤𝜑_∞^𝑛_𝑘=1𝜌_𝑘𝜌_𝑘𝐼−𝐴_𝑘⁻¹,‖𝐹‖≤^𝑛_𝑘=1𝜌_𝑘𝜌_𝑘−𝐴_𝑘; (63)(b)one has 0≤𝐵_𝑗≤𝐴^𝑗1₁⋯𝐴^𝑗𝑛_𝑛,∀𝑗∈𝑍^𝑛₊; (64)(c)one has 0≤𝐵_𝑗≤^𝑛_𝑘=1𝜌^𝑗𝑘_𝑘⁺¹𝜌_𝑘𝐼−𝐴_𝑘⁻¹,𝐵_𝑗≤^𝑛_𝑘=1𝜌^𝑗𝑘_𝑘⁺¹𝜌_𝑘−𝐴_𝑘∀𝑗=𝑗₁,…,𝑗_𝑛∈𝑍^𝑛₊. (65)Then, (b)⇒(a)⇒(c).

Proof. (b)⇒(a). Let 𝐹⊂𝑍^𝑛₊ be a finite subset and let {𝜆_𝑗}_𝑗∈𝐹⊂𝑅 be such that _𝑗∈𝐹𝜆_𝑗𝜑_𝑗≤𝜑=_{𝑗∈𝑍𝑛+}𝑐_𝑗𝜑_𝑗. (66) Then, the order relation on 𝑋 and Cauchy’s inequalities yield 𝜆_𝑗≤Rec_𝑗≤𝑐_𝑗≤𝜑_∞𝜌^𝑗1₁⋯𝜌^𝑗𝑛_𝑛,𝑗=𝑗₁,…,𝑗_𝑛. (67) Using the hypothesis (b) and the previous relations, one obtains _𝑗∈𝐹𝜆_𝑗𝐵_𝑗≤_𝑗∈𝐹+𝜆_𝑗𝐵_𝑗≤_𝑗∈𝐹+𝜆_𝑗𝐴^𝑗1₁⋯𝐴^𝑗𝑛_𝑛≤𝜑_{∞𝑗∈𝑍𝑛+}𝐴₁𝜌₁^𝑗1⋯𝐴_𝑛𝜌_𝑛^𝑗𝑛=𝜑_{∞𝑗1∈𝑍+}𝐴₁𝜌₁^𝑗1⋯_{𝑗𝑛∈𝑍+}𝐴_𝑛𝜌_𝑛^𝑗𝑛=𝜑_∞𝐴𝐼−¹𝜌₁⁻¹⋯𝐴𝐼−^𝑛𝜌_𝑛⁻¹=𝜑_∞^𝑛_𝑘=1𝜌_𝑘𝜌_𝑘𝐼−𝐴_𝑘⁻¹=𝑝𝜑,𝜑∈𝑋. (68) Application of Theorem 3 leads to the existence of an 𝑅-linear positive operator 𝐹 applying 𝑋 into 𝑌 such that 𝐹𝜑_𝑗=𝐵_𝑗,𝑗∈𝑍^𝑛₊,𝐹𝜑𝜑⇒𝐹𝜑𝜑≤𝑝=𝑝−𝜑∀𝜑∈𝑋≤𝑝,𝜑∈𝑋. (69) Using the monotony of the norm on 𝑌, we deduce 𝐹𝜑≤𝜑_∞^∞_𝑘=1𝐴𝐼−^𝑘𝜌_𝑘⁻¹≤𝜑_∞×_{𝑗1∈𝑍+}𝐴₁𝜌₁^𝑗1⋯_{𝑗𝑛∈𝑍+}𝐴_𝑛𝜌_𝑛^𝑗𝑛=𝜑_∞^∞_𝑘=1𝜌_𝑘𝜌_𝑘−𝐴_𝑘,∀𝜑∈𝑋. (70) Thus, the implication (b)⇒(a) is proved.
(a)⇒(c) The hypothesis on positivity of the solution 𝐹 and of functions 𝜑_𝑗 yield 𝐵_𝑗=𝐹𝜑_𝑗≤𝜑_𝑗∞^𝑛_𝑘=1𝜌_𝑘𝜌_𝑘𝐼−𝐴_𝑘⁻¹=^𝑛_𝑘=1𝜌^𝑗𝑘_𝑘⁺¹𝜌_𝑘𝐼−𝐴_𝑘⁻¹⇒𝐵_𝑗=𝐹𝜑_𝑗≤𝜑_𝑗^𝑛_𝑘=1𝜌_𝑘𝜌_𝑘−𝐴_𝑘=^𝑛_𝑘=1𝜌^𝑗𝑘_𝑘⁺¹𝜌_𝑘−𝐴_𝑘,𝑗=𝑗₁,…,𝑗_𝑛∈𝑍^𝑛₊. (71) The proof is complete.

Applying Theorem 16 to the real Hilbert space 𝐻=𝐿²_𝑑𝑡([0,𝜋/2]), 𝑛=2, 𝐴_𝑘, 𝑘=1,2 being the multiplication operator with the functions ℎ₁(𝑡)=sin𝑡, ℎ₂(𝑡)=cos𝑡, one obtains the following.

Corollary 17. Let {𝐵_𝑗;𝑗∈𝑍²₊}⊂𝑌=𝑌(𝐴₁,𝐴₂), 𝜌>1, 𝜌₁=𝜌₂=𝜌.
Consider the following statements: (a)there exists F∈L₊(X,Y) such that 𝐹𝜑_𝑗=𝐵_𝑗,𝑗∈𝑍²₊,₀^𝜋/2𝐹𝜑(ℎ)(𝑡)(𝑡)≤𝜑𝜌⋅ℎ𝑑𝑡_20𝜋/2ℎ²(𝑡)𝜌−sin𝑡𝜌−cos𝑡𝑑𝑡,∀ℎ∈𝐿²_𝑑𝑡𝜋0,2; (72)(b)one has 0≤₀^𝜋/2𝐵_𝑗(ℎ)(𝑡)⋅ℎ(𝑡)𝑑𝑡≤₀^𝜋/2sin^𝑗1𝑡⋅cos^𝑗2𝑡⋅ℎ²(𝑡)𝑗𝑑𝑡,∀ℎ∈𝐻,𝑗=₁,𝑗₂∈𝑍²₊; (73)(c)one has 0≤₀^𝜋/2𝐵_{(𝑗1,𝑗2)}(ℎ)(𝑡)⋅ℎ(𝑡)𝑑𝑡≤𝜌^{𝑗1+𝑗2+2}₀^𝜋/2ℎ²(𝑡)𝜌−sin𝑡𝜌−cos𝑡𝑑𝑡,∀ℎ∈𝐻. (74)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 (𝑢_𝑛)^∞_𝑛=0 in an ordered vector space 𝑌 is said to be nonnegative with respect to the interval 𝐼⊂𝑅 if for any 𝑛∈𝑍₊, we have 𝜆₀+𝜆₁𝑡+⋯+𝜆_𝑛𝑡^𝑛≥0∀𝑡∈𝐼⇒𝜆₀𝑢₀+𝜆₁𝑢₁+⋯+𝜆_𝑛𝑢_𝑛≥0in𝑌. (75) It is clear that for compact intervals and for 𝑌=𝑅, this condition is necessary and sufficient for the existence of a unique positive solution of the moment problem associated to the moments 𝑦_𝑗=𝑢_𝑗∈𝑅, 𝑗∈𝑍₊. The continuity with respect to the supnorm is also obvious. But no information concerning dominating 𝐿¹ norm for the solution holds. Such information would be useful for integral representation of the extension of the solution to the space 𝐿¹(𝐼) (see the next results).

Theorem 19. Let 𝑏∈(0,∞). Consider the following statements:(a)there exists a unique ℎ∈𝐿^∞([0,𝑏]) such that0≤ℎ(𝑡)≤1a.e.,^𝑏₀𝑡^𝑗ℎ(𝑡)𝑑𝑡=𝑦_𝑗,∀𝑗∈𝑍₊; (76)(b)the sequence1,𝑦₀,2𝑦₁,…,𝑛𝑦_𝑛−1[].,…isnonnegativewithrespectto0,𝑏 (77)Then, (b)⇒(a).

Proof. We have ^𝑛_𝑗=0𝜆_𝑗𝜏^𝑗=𝜑₂(𝜏)−𝜑₁(𝜏),𝜑₁,𝜑₂∈𝑋₊,𝑋=𝐿¹([0,𝑏])⇒𝜆₀𝑡+𝜆₁𝑡²2+⋯+𝜆_𝑛𝑡^𝑛+1𝑛+1≤^𝑏₀𝜑₂(𝜏)𝑑𝜏=𝐹₂𝜑₂−𝐹₁𝜑₁,𝐹₁[]⇒≡0,∀𝑡∈0,𝑏_𝑏0𝜑₂𝑑𝜏−𝜆₀𝑡−𝜆₁𝑡²2−⋯−𝜆_𝑛𝑡^𝑛+1[]⇒𝑛+1≥0,∀𝑡∈0,𝑏_𝑏0𝜑₂(𝜏)𝑑𝜏−𝜆₀𝑦₀−𝜆₁2𝑦₁2−⋯−𝜆_𝑛(𝑛+1)𝑦_𝑛⇒𝑛+1≥0_𝑛𝑗=0𝜆_𝑗𝑦_𝑗≤𝐹₂𝜑₂−𝐹₁𝜑₁. (78) Application of Theorem 4, (b)⇒(a), leads to the existence of a linear functional 𝐹 on 𝑋=𝐿¹([0,𝑏]) such that 𝐹𝑥_𝑗=𝑦_𝑗,𝑗∈𝑍₊,0=𝐹₁𝜑≤𝐹𝜑≤𝐹₂𝜑=^𝑏₀𝜑𝑑𝑡,∀𝜑∈𝑋₊. (79) Using the representation of a linear positive functional on 𝐿¹, there exists ℎ∈𝐿^∞𝐹𝜑=([0,𝑏]),0≤ℎ,_𝑏0𝜑⋅ℎ𝑑𝑡,∀𝜑∈𝐿¹([0,𝑏]). (80) From the last equality written for 𝜑=𝜒_𝐵, 𝐵⊂[0,𝑏] being a Borel subset, we infer (via measure theory) that ℎ≤1a.e.,^𝑏₀𝑡^𝑗ℎ(𝑡)𝑑𝑡=𝐹𝑥_𝑗=𝑦_𝑗,∀𝑗∈𝑍₊. (81) The proof is complete.

Lemma 20. Let ℎ∶[𝑎,𝑏]→𝑅 be a positive function in 𝐿¹([𝑎,𝑏]) and 𝑝∶[𝑎,𝑏]→𝑅 a continuous convex function on [𝑎,𝑏]. Then, one has ^𝑏_𝑎𝑝(𝑡)ℎ(𝑡)𝑑𝑡≥^𝑏_𝑎ℎ(𝑡)∫𝑑𝑡⋅𝑝^𝑏𝑎𝑡ℎ(𝑡)𝑑𝑡∫^𝑏_𝑎ℎ(𝑡)𝑑𝑡. (82)

Corollary 21. Let (𝑦_𝑗)_𝑗∈𝑍+ be a sequence of real numbers such that the sequence 1,𝑦₀,2𝑦₁,…,𝑛𝑦_𝑛−1,(𝑛+1)𝑦_𝑛,… (83) is nonnegative with respect to [0,𝑏] and 𝑦₀≠0. Then, 𝑦₀^1−𝑗𝑦^𝑗₁≤𝑦_𝑗≤𝑏^𝑗+1𝑗+1,𝑗∈𝑍₊,𝑦₁≤𝑏⋅𝑦₀. (84)

Proof. Applying Theorem 19 and Lemma 20 to 𝑝(𝑡)=𝑡^𝑗, 𝑡∈[0,𝑏], 𝑗∈𝑍₊, one obtains 𝑦_𝑗=^𝑏₀𝑡^𝑗ℎ(𝑡)𝑑𝑡≤^𝑏₀𝑡^𝑗𝑏𝑑𝑡=^𝑗+1,𝑦𝑗+1_𝑗≥∫^𝑏₀𝑡ℎ(𝑡)𝑑𝑡^𝑗∫^𝑏₀ℎ(𝑡)𝑑𝑡^𝑗−1=𝑦₀^1−𝑗⋅𝑦^𝑗₁,𝑗∈𝑍₊,𝑦₁=^𝑏₀𝑡⋅ℎ(𝑡)𝑑𝑡≤𝑏^𝑏₀ℎ(𝑡)𝑑𝑡=𝑏⋅𝑦₀. (85) 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 𝜓_𝑗(𝑡)=𝑗⋅𝑡^𝑗−1,𝑡∈[0,𝑏],𝑗∈𝐍 (86)

Theorem 22. For a given family of numbers (𝑚_𝑗)^𝑛_𝑗=1, consider the following statements: (a), (b), and (c):(a)there exists ℎ∈𝐿^∞([0,𝑏]) such that0≤ℎ(𝑡)≤1a.e.,𝑚_𝑗=𝑗^𝑏₀𝑡^𝑗−1ℎ(𝑡)𝑑𝑡,𝑗=1,2,…,𝑛; (87)(b)for any family of scalars (𝜆_𝑗)^𝑛_𝑗=0, one has:^𝑛_𝑗=1𝜆_𝑗𝑚_𝑗≤^𝑛_𝑗=1𝜆_𝑗𝑏^𝑗; (88)(c)there exists a Borel subset 𝐸 such that _𝐵𝑗⋅𝑡^𝑗−1𝑑𝑡=𝑚_𝑗,𝑗=1,…,𝑛. (89)Then,(b)⇒(a)⇔(c)

Proof. Let the point (b) be accomplished and assume that ^𝑛_𝑗=1𝜆_𝑗⋅𝑗⋅𝑡^𝑗−1=𝜑₂−𝜑₁,𝜑_𝑗∈𝐿¹([0,𝑏])₊. (90) Then the integration on [0,𝑏] yields ^𝑛_𝑗=1𝜆_𝑗𝑏^𝑗≤^𝑏₀𝜑₂(𝑡)𝑑𝑡=𝐹₂𝜑₂⇒^𝑛_𝑗=1𝜆_𝑗𝑚_𝑗≤^𝑏₀𝜑₂(𝑡)𝑑𝑡. (91) Application of Theorem 4 to 𝐹₁≡0,𝐹₂ defined previously, leads to the existence of a linear positive form 𝐹 on 𝐿¹([0,𝑏]) such that𝐹𝜓_𝑗=𝑚_𝑗𝜓≤,𝑗=1,…,𝑛,𝐹_𝑏0𝜓⋅𝑑𝑡,𝜓∈𝐿¹([0,𝑏])₊⇒𝐹𝜑≤𝐹𝜑⁺+𝐹𝜑⁻≤^𝑏₀𝜑(𝑡)⋅𝑑𝑡,𝜑∈𝐿¹([0,𝑏]). (92)
If characteristic functions of Borel subsets stand for 𝜓, 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 𝑢, namely, [−1,1], is replaced by [0,1], the set of values for ℎ, that stands for the control function 𝑢. The extreme points of the positive part of the unit ball of 𝐿^∞([0,𝑏]) 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 𝑦_1,𝑛<𝑥_1,𝑛<𝑦_2,𝑛<𝑥_2,𝑛<⋯<𝑦_𝑙,𝑛<𝑥_𝑙,𝑛<⋯,𝑛∈𝐍 (93) such that the following relations hold: 𝑚_𝑘=inf_𝑛∈𝐍^∞_𝑗=1𝑥^𝑘_𝑗,𝑛−𝑦^𝑘_𝑗,𝑛,𝑘=1,…,𝑛. (94)

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

Remark 24. In order to approximate the numbers 𝑥^𝑘_𝑗,𝑛,𝑦^𝑘_𝑗,𝑛, one can make use of Fourier approximate expansion of ℎ with respect to the orthonormal sequence attached to the functions 𝑘𝑡^𝑘−1 via Gram-Schmidt algorithm, also using the values of the moments 𝑚_𝑘. Thus, one obtains a smooth approximation ℎ of ℎ, and the intervals of ends 𝑦_𝑙,𝑛,𝑥_𝑙,𝑛 are connected components of the open sets of the form 𝑗_𝑙2^𝑝(𝑛)<ℎ(𝑡)<𝑗_𝑙+12^𝑝(𝑛). (95)

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 (𝑚_{(𝑗1,𝑗2)})_{1≤𝑗1≤𝑛1,1≤𝑗2≤𝑛2} be a given family of real numbers and consider the functions 𝜓_{(𝑗1,𝑗2)}𝑡₁,𝑡₂=𝑗₁𝑗₂𝑡^𝑗1₁⁻¹𝑡^𝑗2₂⁻¹,1≤𝑗_𝑝≤𝑛_𝑝,𝑝=1,2,𝑡₁,𝑡₂∈𝐾₂=0,𝑏₁×0,𝑏₂. (96) Consider the following assertions: (a), (b), (c). Then (b)⇒(a)⇔(c)(a)there exists a Borel function h,0≤h(t₁,t₂)≤1 a.e. such that _𝐾2𝜓_{(𝑗1,𝑗2)}⋅ℎ𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂=𝑚_{(𝑗1,𝑗2)},1≤𝑗₁≤𝑛₁,1≤𝑗₂≤𝑛₂; (97)(b)for any family of scalars {𝜆_(j1,j2)}_{1≤j1≤n11≤j2≤n2}, one has _{1≤𝑗1≤𝑛11≤𝑗2≤𝑛2}𝜆_{(𝑗1,𝑗2)}𝑚_{(𝑗1,𝑗2)}≤_{1≤𝑗1≤𝑛11≤𝑗2≤𝑛2}𝜆_{12)(𝑗,𝑗}𝑏^𝑗1₁𝑏^𝑗2₂; (98)(c)there exists a Borel subset B₂⊂K₂ such that _𝐵2𝜓_{(𝑗1,𝑗2)}𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂=𝑚_{(𝑗1,𝑗2)},1≤𝑗_𝑝≤𝑛_𝑝,𝑝=1,2. (99)

Corollary 27. If one of the conditions (a), (c) from Theorem 26 is accomplished, there are sequences 𝑦_1,𝑛<𝑥_1,𝑛<⋯<𝑦_𝑙,𝑛<𝑥_𝑙,𝑛𝑣<⋯,_1,𝑛<𝑢_1,𝑛<⋯<𝑣_𝑙,𝑛<𝑢_𝑙,𝑛<⋯,𝑛∈𝐍, (100) such that 𝑚_{(𝑘1,𝑘2)}=inf_{𝑛∈𝐍𝑙∈𝑁}𝑥^𝑘1_𝑙,𝑛−𝑦^𝑘1_𝑙,𝑛⋅𝑢^𝑘2_𝑙,𝑛−𝑣^𝑘2_𝑙,𝑛. (101)

Proof. The Borel subset 𝐵₂ is the joint of a 𝐺_𝛿 set and a null set. For an open subset 𝐷_𝑛⊃𝐵₂, we consider its decomposition into cells [28], used in the construction of Lebesgue measure. If we determine smooth approximations ℎ of ℎ by means of Gram-Schmidt algorithm for the functions 𝜓_{(𝑗1,𝑗2)} and the given moments 𝑚_{(𝑗1,𝑗2)} via Fourier expansion, then the sequences from the present corollary can be determined by means of cell decomposition of the open subsets 𝑡₁,𝑡₂;𝑚(𝑙)2^𝑝(𝑛)<ℎ𝑡₁,𝑡₂<𝑚(𝑙)+12^𝑝(𝑛). (102) 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 𝑦_𝑙,𝑛,𝑥_𝑙,𝑛,𝑣_𝑙,𝑛,𝑢_𝑙,𝑛. 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 𝑛 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 (𝑚_𝑘)_𝑘≥1 be a sequence of real numbers. Consider the following assertions:(a)there exists a Borel function h such that 0≤ℎ(𝑡)≤1a.e.,𝑚_𝑘=𝑘^𝑏_𝑎𝑡^𝑘−1ℎ(𝑡)𝑑𝑡,𝑘∈𝐍⧵{0}; (103)(b)for any natural number n≥1, and any 𝜀>0, there exists nonnegative scalars 𝛽_j,j=1,…,n, and sequences: 𝑦_𝑗,1<𝑥_𝑗,1<⋯<𝑦_𝑗,𝑙<𝑥_𝑗,𝑙<⋯,𝑗=1,…,𝑛 (104) such as 1−𝜀≤^𝑛_𝑗=1𝛽_𝑗≤1,𝑚_𝑘=lim_𝑛^𝑛_𝑗=1𝛽_𝑗𝑙𝑥^𝑘_𝑗,𝑙−𝑦^𝑘_𝑗,𝑙; (105)(c)for any n∈𝐍⧵{0}, there exists a Borel subset B_n such that 𝑚_𝑘=𝑘_𝐵𝑛𝑡^𝑘−1𝑑𝑡,𝑘=1,…,𝑛; (106)(d)for any natural n≥1 and any {𝜆₁,…,𝜆_n}⊂R, the following relation holds true: ^𝑛_𝑘=1𝜆_𝑘𝑚_𝑘≤^𝑛_𝑘=1𝜆_𝑘𝑏^𝑘−𝑎^𝑘. (107)

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 𝐿^∞([𝑎,𝑏]) with the positive cone of the same space. Then, we must have ℎ=lim_𝑛^𝑛_𝑗=1𝛼_𝑗𝜒_𝐵𝑗=lim_𝑛^𝑛_𝑗=1𝛽_𝑗𝜒_𝐷𝑗,𝛼_𝑗≥0,^𝑛_𝑗=1𝛼_𝑗𝛽=1,_𝑗=𝑚𝐵_𝑗𝑚𝐷_𝑗⋅𝛼_𝑗∈(1−𝜀)𝛼_𝑗,𝛼_𝑗,𝐵_𝑗⊂𝐷_𝑗,𝑗=1,…,𝑛,𝐷_𝑗 (108) which are suitably chosen open subsets, where the limit is in the weak topology on 𝐿^∞, with respect to the dual pair (𝐿¹,𝐿^∞);  𝑚 is the Lebesgue measure. This leads to 𝑚_𝑘=𝑘^𝑏_𝑎𝑡^𝑘−1ℎ(𝑡)𝑑𝑡=lim_𝑛^𝑛_𝑗=1𝛽_𝑗𝐷𝑗𝑘𝑡^𝑘−1𝑑𝑡. (109) Since each open subset 𝐷_𝑗 has an at most countable decomposition, 𝐷_𝑗=_𝑙𝑦_𝑗,𝑙,𝑥_𝑗,𝑙, (110) the conclusion (b) follows. For the converse implication, observe that each step function, ℎ_𝑛∶=^𝑛_𝑗=1𝛽_𝑗𝜒_𝐷𝑗,𝐷_𝑗=_𝑙𝑦_𝑗,𝑙,𝑥_𝑗,𝑙𝛽,𝑗=1,…,𝑛′_𝑗≥0,1−𝜀≤^𝑛_𝑗=1𝛽_𝑗≤1, (111) is an element of the positive part of the unit ball in 𝐿^∞, and this subset is weakly compact. Let ℎ be the (𝑤)-limit of a subsequence of the sequence (ℎ_𝑛)_𝑛. Then, we have 𝑚_𝑘=lim_𝑛^𝑏_𝑎ℎ_𝑚𝑛(𝑡)𝑘⋅𝑡^𝑘−1𝑑𝑡=𝑘^𝑏_𝑎ℎ(𝑡)⋅𝑡^𝑘−1𝑑𝑡, (112) 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 𝐽₀⊂𝐍⧵{0} is a finite subset and {𝜆_𝑗;𝑗∈𝐽₀}⊂𝑅, then the following implications hold true: _𝑗∈𝐽0𝜆_𝑗⋅𝑗⋅𝑡^𝑗−1=𝜑₂−𝜑₁,𝜑_𝑝∈𝐿¹₊⇒_𝑗∈𝐽0𝜆_𝑗⋅𝑏^𝑗−𝑎^𝑗≤^𝑏_𝑎𝜑₂𝑑𝑡=𝐹₂𝜑₂−𝐹₁𝜑₁,𝐹₁≡0⇒_𝑗∈𝐽0𝜆_𝑗⋅𝑚_𝑗≤^𝑏_𝑎𝜑₂𝑑𝑡=𝐹₂𝜑₂. (113)
Application of Theorem 4 leads to the existence of a linear functional 𝐹 on 𝐿¹([𝑎,𝑏]), verifying 𝐹𝑘𝑡^𝑘−1=𝑚_𝑘𝜓≤,𝑘≥1,𝑘∈𝐍,0≤𝐹^𝑏_𝑎𝜓𝑑𝑡,∀𝜓∈𝐿¹₊([𝑎,𝑏]). (114) Now, the conclusion follows by measure theory [28].

Remark 29. For the full moment problem, the following algorithm holds in determining 𝑦_𝑗,𝑙,𝑥_𝑗,𝑙.
Step  1. Approximating the function ℎ.
Let (𝑒_𝑛)_𝑛≥1 be a Hilbert base constructed by the aid of Gram-Schmidt procedure, applied to the system of linearly independent functions 𝜑_𝑛(𝑡)=𝑛⋅𝑡^𝑛−1,𝑛∈𝐍⧵{0}. (115) Then, for each fixed 𝑛≥1, one has 𝑒_𝑛=^𝑛_𝑗=1𝑎_𝑗^(𝑛)𝜑_𝑗⇒⟨ℎ,𝑒_𝑛⟩=^𝑛_𝑗=1𝑎_𝑗^(𝑛)⟨ℎ,𝜑_𝑗⟩=^𝑛_𝑗=1𝑎_𝑗^(𝑛)𝑚_𝑗, (116) where the coefficients 𝑎_𝑗^(𝑛) are known from the Gram-Schmidt procedure. Hence, we can determine each Fourier coefficient of ℎ that is, we can approximate ℎ in 𝐿²-norm by a sequence of polynomial functions ℎ_𝑛, 𝑛≥1. Then, there exists a subsequence [28] ℎ_𝑘𝑛→ℎ pointwise almost everywhere in [𝑎,𝑏].
Step  2. For each 𝑛∈𝐍⧵{0}, the subsets, 𝑚𝑡;^𝑙2^𝑝<ℎ_𝑘𝑛(𝑡)<𝑚_𝑙+12^𝑝,𝑝,𝑚_𝑙∈𝐍, (117) are open subsets. Their connected components have end points approximations of the unknowns 𝑦_𝑗,𝑙, 𝑥_𝑗,𝑙. From [26], using a weakly compact standard argument, we can obtain ℎ as the limit of a subsequence of (𝜒_𝐵𝑛)_𝑛, where 𝐵_𝑛 are as at point (c) of Theorem 28. Considering a suitable open set 𝐷_𝑘𝑛⊃𝐵_𝑘𝑛, from (a) one obtains:𝑚_𝑘≈_𝐷𝑘𝑛𝑘⋅𝑡^𝑘−1𝑑𝑡≈_𝑙𝑥^𝑘_𝑛,𝑙−𝑦^𝑘_𝑛,𝑙,𝐷_𝑘𝑛=_𝑙𝑦_𝑛,𝑙,𝑥_𝑛,𝑙. (118)

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

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

Theorem 30. Let 𝑋 be a locally convex space, 𝑌 an order-complete vector lattice with strong order unit 𝑢₀, and 𝑆⊂𝑋 a vector subspace. Let 𝐴⊂𝑋 be a convex subset with the following qualities:(i)there exists a neighborhood V of the origin such that (S+V)∩A=Φ (A and S are distanced);(ii)A is bounded.
Then, for any equicontinuous family of linear operators {𝑓_𝑗}_𝑗∈𝐽⊂𝐿(𝑆,𝑌) and for any 𝑦∈𝑌₊⧵{0}, there exists an equicontinuous family {𝐹_𝑗}_𝑗∈𝐽⊂𝐿(𝑋,𝑌) such that𝐹_𝑗∣_𝑆=𝑓_𝑗,𝐹_𝑗∣_𝐴≥𝑦,∀𝑗∈𝐽. (119)Moreover, if 𝑉 is a neighborhood of the origin such that 𝑓_𝑗(𝑉∩𝑆)⊂−𝑢₀,𝑢₀,(𝑆+𝑉)∩𝐴=Φ,0<𝛼∈𝑅s.t.𝑝_𝑉|_𝐴≤𝛼,𝛼₁>0s.t.𝑦≤𝛼₁𝑢₀, (120)then the following relations hold:𝐹_𝑗(𝑥)≤1+𝛼+𝛼₁𝑝_𝑉(𝑥)⋅𝑢₀,𝑥∈𝑋,𝑗∈𝐽. (121)

The last relation of Theorem 30 gives a relationship between an upper bound and the lower bound of 𝐹_𝑗,𝑗∈𝐽. The next result is a consequence of the preceding one, in terms of the moment problem. The topology on 𝑌 is the norm topology for which the unit closed ball is the interval [−𝑢₀,𝑢₀].

Theorem 31. Let ,𝑥0<𝑏∈𝑅,𝑋=𝐶([0,𝑏])_𝑗(𝑡)=𝑡^𝑗,𝑗∈𝑍₊[],𝜑,𝑗≥1,𝑡∈0,𝑏_{𝑘𝑘∈𝑍+}⊂𝑋,𝜑_𝑘≤1,𝜑₀≡1,𝜑_𝑘(0)=1,𝑘∈𝑍₊. (122)
Let 𝑌 be an order-complete vector lattice with strong order unit 𝑢₀and (𝑦_𝑛)_𝑛≥1 a sequence in 𝑌 such that (𝑢₀,𝑦₁,𝑦₂,…) is positive with respect to [0,𝑏]. Then, for any 𝛼₁∈𝑅₊, there exists 𝐹∈𝐿(𝑋,𝑌) such that 𝐹𝑥_𝑗=𝑦_𝑗,𝑗∈𝑍₊𝐹𝜑,𝑗≥1,_𝑘≥𝛼₁𝑢₀,𝑘∈𝑍₊,𝐹(𝑥)≤2+𝛼₁⋅‖𝑥‖⋅𝑢₀,𝑥∈𝑋. (123) Moreover, if 𝛼₁≥1 and 𝑌 is endowed with a linear topology such that its positive cone is closed and normal, then 𝐹 is continuous and positive.

Proof. Let 𝑆=𝑆𝑝𝑥_𝑗;𝑗∈𝑍₊,𝑗≥1,𝐴=co𝜑_𝑘;𝑘∈𝑍₊. (124) For all 𝑝∈𝑆,𝑎∈𝐴, we have 𝑝−𝑎≥𝑝(0)−𝑎(0)=1, (125) so that 𝑑(𝑆,𝐴)≥1 holds. This leads to (𝑆+𝐵(0,1))∩𝐴=Φ, so we can take in Theorem 30  𝑉=𝐵(0,1),𝑝_𝑉=‖⋅‖_∞. From the hypothesis on ‖𝜑_𝑘‖, 𝑘∈𝑍₊, we infer that𝜑_𝑘≤1∀𝑘∈𝑍₊⇒‖⋅‖|_𝐴≤1⇒𝛼∶=1, (126) where 𝛼 is the constant from Theorem 30. We define that 𝑦∶=𝛼₁⋅𝑢₀,𝑓^𝑛_𝑗=1𝜆_𝑗𝑥_𝑗∶=^𝑛_𝑗=1𝜆_𝑗𝑦_𝑗. (127)
Let ∑^𝑛_𝑗=1𝜆_𝑗𝑥_𝑗∈𝑆∩𝐵(0,1). Using the hypothesis on the sequence which is positive with respect to [0,𝑏], this yields sup^𝑛_𝑗=1𝜆_𝑗𝑡^𝑗[]⇒;𝑡∈0,𝑏<1_𝑛𝑗=1𝜆_𝑗𝑡^𝑗+1>0,1−^𝑛_𝑗=1𝜆_𝑗𝑡^𝑗>0⇒𝑢₀+^𝑛_𝑗=1𝜆_𝑗𝑦_𝑗≥0,𝑢₀−^𝑛_𝑗=1𝜆_𝑗𝑦_𝑗≥0⇒𝑓^𝑛_𝑗=1𝜆_𝑗𝑥_𝑗=^𝑛_𝑗=1𝜆_𝑗𝑦_𝑗∈−𝑢₀,𝑢₀⇒𝑓(𝑆∩𝐵(0,1))⊂−𝑢₀,𝑢₀. (128)
The conclusion is that 𝑓 verifies the conditions in the statement of Theorem 30. Application of the latter theorem leads to the existence of an extension 𝐹 of 𝑓 such that 𝐹|_𝐴≥𝛼₁𝑦₀,𝐹(𝑥)≤2+𝛼₁⋅‖𝑥‖⋅𝑢₀,𝑥∈𝑋. (129)
Whence, the first part of the theorem is proved. It remains to prove the continuity and positivity of the extension 𝐹, under the hypothesis mentioned in the statement. Replacing 𝑥 by −𝑥, it is easy to see that |𝐹(𝑥)|≤2+𝛼₁⋅‖𝑥‖⋅𝑢₀,𝑥∈𝐶([0,𝑏]). (130) From these relations, also using that the positive cone 𝑌₊ is normal, the continuity of 𝐹 follows. In order to prove that 𝐹 is positive, it is sufficient to show that 𝐹(𝑝)≥0 for all positive polynomials 𝑝. Hypothesis yields 𝑝(𝑡)=𝜆₀+^𝑛_𝑗=1𝜆_𝑗𝑡^𝑗≥0∀𝑡∈[0,𝑏]⇒𝜆₀𝑢₀+^𝑛_𝑗=1𝜆_𝑗𝑦_𝑗𝑝≥0⇒𝐹=𝜆₀𝐹𝜑₀+^𝑛_𝑗=1𝜆_𝑗𝑦_𝑗≥𝜆₀𝛼₁⋅𝑢₀−𝑢₀=𝜆₀𝛼₁−1⋅𝑢₀≥0. (131) The proof is complete.

A consequence of Theorem 31 is the following one.

Corollary 32. Let 𝐻 be a Hilbert space, 𝐴∈𝐴(𝐻) a positive selfadjoint operator, and 𝑌=𝑌(𝐴) the commutative algebra of selfadjoint operators defined in Section 2. Let 𝑏∈𝑅₊ be such that the spectrum 𝑆(𝐴)⊂[0,𝑏] and 𝛼₁≥1. Then, there exists a nondecreasing mapping 𝜎∶[0,𝑏]→𝑌 such that ^𝑏₀𝑡^𝑗𝑑𝜎(𝑡)=𝐴^𝑗,𝑗∈𝑍₊,𝑗≥1,_𝑏0𝑒^−𝑘𝑡𝑑𝜎(𝑡)≥𝛼₁𝐼,𝑘∈𝑍₊,_𝑏0𝑥(𝑡)𝑑𝜎(𝑡)≤2+𝛼₁⋅‖𝑥‖⋅𝐼,𝑥∈𝐶([0,𝑏]). (132) In particular, for such a mapping 𝜎, the following relation holds: 𝛼₁+1≤‖𝜎(𝑏)−𝜎(0)‖+exp(−𝑘𝐴),𝑘∈𝑍₊. (133)

Proof. We apply Theorem 31 to 𝜑_𝑘(𝑡)=𝑒^−𝑘𝑡, 𝑘∈𝑍₊, 𝑡∈[0,𝑏], 𝑢₀=𝐼. The sequence (𝐼,𝐴,𝐴²,…,𝐴^𝑛,…) is positive with respect to [0,𝑏] because of the positivity of the spectral measure attached to 𝐴. Due to the hypothesis of Theorem 31 being accomplished, there exists a positive continuous solution 𝐹 satisfying the relations in the statement of the latter theorem. This solution applies norm-bounded subsets into order-bounded subsets of 𝑌(𝐴). Application of [14, page 272], yields the existence of a “representing” nondecreasing mapping 𝜎, such that the vector measure 𝑑𝜎 represents 𝐹. Next, we prove the last relation of the statement: 𝛼₁𝐼≤^𝑏₀𝑒^−𝑘𝑡𝑑𝜎=𝜎(𝑏)−𝜎(0)+^∞_𝑚=1(−𝑘)^𝑚𝐴𝑚!^𝑚=𝜎(𝑏)−𝜎(0)−𝐼+exp(−𝑘𝐴). (134) Further computation yields 𝛼₁+1𝐼≤𝜎(𝑏)−𝜎(0)+exp(−𝑘𝐴)⇒𝛼₁≤‖𝜎(𝑏)(0+(+1−𝜎)‖exp−𝑘𝐴),𝑘∈𝑍₊. (135) The proof is complete.

Application of Theorem 31 to positive sequences of real numbers with respect to [0,𝑏], the functions 𝜑_𝑘, 𝑘∈𝑍₊ being as in Corollary 32, yields the following consequence, which we state without proof.

Corollary 33 (see [5, Corollary 3.3]). Let (1,𝑦₁,𝑦₂,…) be a sequence of real numbers, which is positive with respect to [0,𝑏], 𝑏>0 being arbitrary. Then, the following inequalities hold: 𝑦₁≤3^{(𝑗−1)/𝑗}⋅𝑦_𝑗^1/𝑗,𝑗∈𝑍₊,𝑗≥1. (136) In particular, one has: 𝑦₁≤3liminf𝑦_𝑗^1/𝑗.

Next, we consider the analogue of Theorem 31, for a space 𝑋 of analytic functions. Let 𝑏>1 and 𝑋 be the space of all functions 𝜑 which can be represented as an absolutely convergent series, 𝜑(𝑧)=^∞_𝑗=0𝜆_𝑗𝑧^𝑗,|𝑧|<𝑏,𝜆_𝑗∈𝑅,𝜑 (137) being continuous in the closed disc |𝑧|≤𝑏. Let 𝜑_𝑗(𝑧)=𝑧^𝑗, 𝑗∈𝑍₊, 𝑌=𝐿^∞(Ω),(Ω,𝜇) being a measurable space, and 𝑢₀(𝜔)=1,for all 𝜔∈Ω.

Theorem 34. Let{𝜓_𝑘}_𝑘∈𝑍+⊂𝑋 be a sequence such that 𝜓_𝑘∞≤𝑀,𝜓_𝑘(0)=1,∀𝑘∈𝑍₊. (138) Assume that 𝑏>1 and let (𝑦_𝑗)_𝑗≥1 be a sequence in 𝑌 such that 𝑦_𝑗≤𝑏−1,𝑗≥1,𝑗∈𝑍. (139) Then, for any 𝑦∈𝑌₊⧵{0}, there exists 𝐹∈𝐿(𝑋,𝑌) such that 𝐹𝜑_𝑗=𝑦_𝑗,𝑗≥1,𝐹𝜓_𝑘≥𝑦,𝑘∈𝑍₊,𝐹(𝑥)≤1+𝑀+𝑦_∞⋅‖𝑥‖_∞⋅𝑢₀,𝑥∈𝑋. (140)

Proof. If 𝑆=𝑆𝑝{𝜑_𝑗;𝑗≥1},𝐴=co{𝜓_𝑘;𝑘∈𝑍₊}, we have 𝑑()((()𝑆,𝐴≥1,𝑆+𝐵0,1))∩𝐴=Φ,𝑉=𝐵0,1,𝑝_𝑉=‖⋅‖_∞. (141) Let 𝜑∈𝐵(0,1)∩𝑆, 𝜑=∑^𝑛_𝑗=1𝜆_𝑗𝜑_𝑗, ‖𝜑‖<1. Using the Cauchy inequalities, we derive 𝜆_𝑗≤𝜑𝑏^𝑗<1𝑏^𝑗,𝑗∈{1,2,…,𝑛}. (142) Define 𝑓(∑^𝑛_𝑗=1𝜆_𝑗𝜑_𝑗)=∑^𝑛_𝑗=1𝜆_𝑗𝑦_𝑗, 𝑛∈𝑍, 𝑛≥1, 𝜆_𝑗∈𝑅. Using also the hypothesis on ‖𝑦_𝑗‖,𝑗≥1, one obtains ^𝑛_𝑗=1𝜆_𝑗𝑦_𝑗≤^𝑛_𝑗=1𝜆_𝑗⋅𝑦_𝑗≤^𝑛_𝑗=11𝑏^𝑗⋅𝑦_𝑗∞⋅𝑢₀≤^∞_𝑗=11𝑏^𝑗⋅(𝑏−1)𝑢₀=𝑢₀. (143) Whence, 𝜑=∑^𝑛_𝑗=1𝜆_𝑗𝜑_𝑗∈𝑆∩𝐵(0,1)⇒𝑓(𝜑)∈[−𝑢₀,𝑢₀]. Since ‖⋅‖|_𝐴≤𝑀,𝑦≤𝑦_∞⋅𝑢₀, (144) one takes in Theorem 30  𝛼=𝑀, 𝛼₁=‖𝑦‖_∞. 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 𝑅^𝑛, the form of positive polynomials is not necessarily used.

Theorem 35. Let (𝑇,𝜈) be a measurable space, where 𝜈 is a positive 𝜎-finite measure on 𝑇. Let 𝑋=𝐿¹_𝜈(𝑇), {𝑥_𝑗}_𝑗∈𝐽⊂𝑋, {𝑦_𝑗}_𝑗∈𝐽⊂𝑅. Consider the following statements:(a)there exists h∈L^∞_𝜈(T) such that _𝑇𝑥_𝑗(𝑡)⋅ℎ(𝑡)𝑑𝜈=𝑦_𝑗(𝑡),𝑗∈𝐽,−1≤ℎ≤1𝜈-𝑎.𝑒.; (145)(b)for any finite subset J₀⊂J and any {𝜆_j;j∈J₀}⊂R, one has _{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝑦_𝑖𝑦_𝑗≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝑇𝑥_𝑖(𝑡)𝑑𝜈⋅_𝑇𝑥_𝑗(𝑡)𝑑𝜈; (146)(c)for any finite subset J₀⊂J and any {𝜆_j;j∈J₀}⊂R, one has _{𝑖,𝑗∈𝐽0}𝜆_𝑖𝜆_𝑗𝑦_𝑖𝑦_𝑗≤_{𝑖,𝑗∈𝐽0}𝜆_𝑖⋅𝜆_𝑗⋅_𝑇𝑥_𝑖(𝑡)𝑑𝜈⋅_𝑇𝑥_𝑗(𝑡)𝑑𝜈. (147)
Then, (b)⇒(a)⇒(c).

Proof. (b)⇒(a). The following implications hold:_𝑗∈𝐽0𝜆_𝑗𝑥_𝑗=𝜑₂−𝜑₁,𝜑_𝑘∈𝑋₊,𝑘=1,2⇒−_𝑇𝜑₁𝑑𝜈−_𝑇𝜑₂𝑑𝜈≤_𝑇𝜑₂𝑑𝜈−_𝑇𝜑₁=𝑑𝜈_𝑗∈𝐽0𝜆_𝑗𝑇𝑥_𝑗𝑑𝜈≤_𝑇𝜑₂𝑑𝜈+_𝑇𝜑₁=𝑑𝜈_𝑇𝜑₂𝑑𝜈−−_𝑇𝜑₁𝑑𝜈=𝐹₂𝜑₂−𝐹₁𝜑₁⇒_𝑗∈𝐽0𝜆_𝑗𝑇𝑥_𝑗𝑑𝜈≤𝐹₂𝜑₂−𝐹₁𝜑₁. (148) On the other hand, the condition (b) can be rewritten as _𝑗∈𝐽0𝜆_𝑗𝑦_𝑗²≤_𝑗∈𝐽0𝜆_𝑗𝑇𝑥_𝑗𝑑𝜈²⟺_𝑗∈𝐽0𝜆_𝑗𝑦_𝑗≤_𝑗∈𝐽0𝜆_𝑗𝑇𝑥_𝑗𝑑𝜈. (149) From the preceding relation, we conclude _𝑗∈𝐽0𝜆_𝑗𝑦_𝑗≤_𝑗∈𝐽0𝜆_𝑗𝑦_𝑗≤_𝑗∈𝐽0𝜆_𝑗𝑇𝑥_𝑗(𝑡)𝑑𝜈≤𝐹₂𝜑₂−𝐹₁𝜑₁. (150) Whence, the conditions from the statement of Theorem 4 are accomplished, so that there exists a linear form 𝐹 on 𝑋 such that −_𝑇𝜑𝑑𝜈=𝐹₁𝜑≤𝐹𝜑≤𝐹₂𝜑=_𝑇𝜑𝑑𝜈⇒𝐹𝜑≤_𝑇𝜑𝑑𝜈∀𝜑∈𝑋₊⇒𝐹𝜑≤𝐹𝜑⁺+𝐹𝜑⁻≤_𝑇𝜑⁺+𝜑⁻𝑑𝜈=_𝑇𝜑𝑑𝜈=𝜑₁⇒‖𝐹‖≤1. (151) Following the representation of continuous linear functionals on 𝐿¹ spaces, there exist ℎ∈𝐿^∞_𝜈(𝑇),𝐹𝜑=_𝑇𝜑(𝑡)ℎ(𝑡)𝑑𝜈∀𝜑∈𝐿¹_𝜈(𝑇),‖ℎ‖_∞=‖𝐹‖≤1⇒−1≤ℎ(𝑡)≤1𝜈-a.e. (152) We also infer that 𝑦_𝑗=𝐹𝑥_𝑗=_𝑇𝑥_𝑗(𝑡)ℎ(𝑡)𝑑𝜈,∀𝑗∈𝐽. (153) Thus, the proof of the implication (b)⇒(a) is finished. The implication (a)⇒(c) is almost obvious: _𝑗∈𝐽0𝜆_𝑗𝑦_𝑗=_𝑗∈𝐽0𝜆_𝑗𝑇𝑥_𝑗⋅ℎ⋅𝑑𝜈≤_𝑗∈𝐽0𝜆_𝑗⋅_𝑇𝑥_𝑗(𝑡)⋅|ℎ(𝑡)|⋅𝑑𝜈≤_𝑗∈𝐽0𝜆_𝑗⋅_𝑇𝑥_𝑗(𝑡)⋅𝑑𝜈. (154) Taking the squares of the two members, the inequality is preserved. The proof is complete.

Corollary 36. Let 𝑇=𝑡₁,𝑡₂∈𝑅²;0≤𝑡₁<∞,0≤𝑡₂≤exp−𝑡₁,𝑦_{(𝑗1,𝑗2)(𝑗1,𝑗2)∈𝑍2+}⊂𝑅. (155)Consider the following statements: (a)there exists a Lebesgue measurable function ℎ on 𝑇 such that _𝑇𝑡^𝑗1₁𝑡^𝑗2₂ℎ𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂=𝑦_{(𝑗1,𝑗2)},∀𝑗₁,𝑗₂∈𝑍²₊,−1≤ℎ𝑡₁,𝑡₂≤1a.e.; (156)(b)for any finite subset J₀⊂Z²₊ and any {𝜆_(j1,j2);(j₁,j₂)∈J₀}⊂R, one has _{(𝑖1,𝑖2),(𝑗1,𝑗2)∈𝐽0}𝜆_{(𝑖1,𝑖2)}𝜆_{(𝑗1,𝑗2)}𝑦_{(𝑖1,𝑖2)}𝑦_{(𝑗1,𝑗2)}≤_{(𝑖1,𝑖2),(𝑗1,𝑗2)∈𝐽0}𝜆_{(𝑖1,𝑖2)}𝜆_{(𝑗1,𝑗2)}𝑖₁!𝑖₂+1^𝑖1+2⋅𝑗₁!𝑗₂+1^𝑗1+2. (157)
Then, (b)⇒(a).

Note that 𝑇 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.