Markov Moment Problems on Special Closed Subsets of ${\mathbb{R}}^{\mathit{n}}$

Abstract

First, this paper provides characterizing the existence and uniqueness of the linear operator solution $T$ for large classes of full Markov moment problems on closed subsets $F$ of ${\mathbb{R}}^n.$ One uses approximation by special nonnegative polynomials. The case when $F$ is compact is studied. Then the cases when $F={\mathbb{R}}^n$ and $F={\mathbb{R}}_{+}^n$ are under attention. Here, the main findings consist in proving and applying the density of special polynomials, which are sums of squares, in the positive cone of $L_{\nu }^1\left({\mathbb{R}}^n\right),$ and respectively of $L_{\nu }^1\left({\mathbb{R}}_{+}^n\right),$ for a large class of measures $\nu$. One solves the important difficulty created by the fact that on ${\mathbb{R}}^n, n\ge 2,$ there exist nonnegative polynomials which are not expressible in terms of sums of squares. This is the second aim of the paper. On the other hand, two types of symmetry are outlined. Both these symmetry properties appear naturally from the thematic mentioned above. This is the third aim of the paper. They lead to new statements, illustrated in corollaries, and supported by a few examples.

1. Introduction

This work studies main aspects of the Markov moment problem on closed bounded or on closed unbounded subsets of ${\mathbb{R}}^n$. The first aim is to solve this problem for a large class of compact subsets, involving solutions having as codomain an order complete Banach lattice of self-adjoint operators. Most of the results involve linear operators $T$ from $L_{\nu }^1\left(F\right)$ into an order complete Banach lattice $Y,$ where $\nu$ is a moment determinate measure on the closed subset $F\subseteq {\mathbb{R}}^n$. We require the order completeness of $Y$ to make valid Hahn-Banach extension theorems for classes of linear operators having $Y$ as codomain. Recall that a positive regular Borel measure $\nu$ on the closed subset $F$ is called moment determinate if it is uniquely determined by its moments ${\int }_F{t}^{j}d\nu , j\in {\mathbb{N}}^n,$ which are assumed to be finite numbers. In the sequel, we review basic usual notations. Namely, we denote:

$$\mathbb{N}=\left\{0,1,2,\dots \right\}, {\mathbb{R}}_{+}=\left[0,+\infty \right),$$

$${\phi }_j\left(t\right)=t^j=t_1^{j_1}\cdots t_n^{j_n}, j=\left(j_1,\dots ,j_n\right)\in {\mathbb{N}}^n, t=\left(t_1,\dots ,t_n\right)\in F, n\in \mathbb{N}, n\ge 1,$$

$\mathcal{P}\left(F\right)$ is the vector space of all polynomials with real coefficients in variable

$$t=\left(t_1,\dots ,t_n\right)\in F, {\mathcal{P}}_{+}\left(F\right)=\left\{p\in \mathcal{P}\left(F\right);p\left(t\right)\ge 0 \forall t\in F\right\},{\mathcal{P}}_{++}\left(F\right)\subseteq {\mathcal{P}}_{+}\left(F\right)$$

is a convex cone formed by finite sums of special polynomials involving squares multiplied by nonnegative polynomials defining the set $F.$ If $p_1,\dots ,p_n$ are nonnegative polynomials on $\mathbb{R}$ (respectively on ${\mathbb{R}}_{+}$), we denote

$$({\mathrm{p}}_1\otimes \cdots \otimes {\mathrm{p}}_{\mathrm{n}})\left({\mathrm{t}}_1,\dots ,{\mathrm{t}}_{\mathrm{n}}\right)={\mathrm{p}}_1\left({\mathrm{t}}_1\right)\cdots {\mathrm{p}}_{\mathrm{n}}\left({\mathrm{t}}_{\mathrm{n}}\right), t=\left(t_1,\dots .,t_n\right)\in {\mathbb{R}}^n.$$

(1)

This is a special nonnegative polynomial on ${\mathbb{R}}^n$, which is a sum of squares, since so is each $p_l\in {\mathcal{P}}_{+}\left(\mathbb{R}\right)$. We recall that $p\in {\mathcal{P}}_{+}\left(\mathbb{R}\right)\iff p\left(t\right)=q^2\left(t\right)+r^2\left(t\right) \forall t\in \mathbb{R}$ for some $q,r\in \mathbb{R}\left[t\right]$ (cf. [1]). Similarly, $p\in {\mathcal{P}}_{+}\left({\mathbb{R}}_{+}\right)\iff p\left(t\right)=q^2\left(t\right)+t{r}^2\left(t\right) \forall t\in {\mathbb{R}}_{+}$ for some polynomials $q,r\in \mathbb{R}\left[t\right]$. Therefore, each tensor product (1) with $p_l\in {\mathcal{P}}_{+}\left({\mathbb{R}}_{+}\right), l=1,\dots ,n$ is expressible in terms of a sums of squares multiplied by $t_1^{e_1}\cdots t_n^{e_n}, t_l\in {\mathbb{R}}_{+}, e_l\in \left\{0,1\right\}, l=1,\dots ,n.$ The classical moment problem and the methods used in solving it are part of the general framework of functional analysis and polynomial approximation on unbounded subsets. In the proof of such approximation results on unbounded closed subsets, basic theorems of functional analysis and measure theory as well as sufficient criterions for determinacy are applied. In the sequel, we review the formulation of the classical moment problem and of the Markov moment problem, also reminding their vector valued variants. One denotes by $C_c\left(F\right)$ the vector space of all real valued continuous compactly supported functions defined on $F.$ For general results, terminology, and related background, see paragraphs from books and monographs [1,2,3,4,5,6,7,8,9,10]. Given a sequence ${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$ of real numbers, then one studies the existence, uniqueness, and construction of a linear positive form $T_1$ defined on a function space $X_1$ containing polynomials and continuous compactly supported real functions, such that the moment conditions

$${\mathrm{T}}_1\left({\mathsf{\phi }}_{\mathrm{j}}\right)={\mathrm{y}}_{\mathrm{j}}, \mathrm{j}\in {\mathbb{N}}^{\mathrm{n}},$$

(2)

are satisfied. Usually, if $F\subseteq {\mathbb{R}}^n$ is a closed unbounded subset, and $\nu$ is a positive regular measure on $F,$ with finite moments of all orders, we put $X=L_{\nu }^p\left(F\right),$ where $1\le p<+\infty$ and $X_1=\left\{f\in X;\exists p\in \mathcal{P}, \left|f\right|\le p\right\}$. The next step is to prove that $T_1$ is continuous on $X_1.$ Assuming this is done, since $C_c\left(F\right)$ is contained in $X_1$ and is dense in $X$, $T_1$ has a unique positive linear continuous extension $T$ defined on the entire space $X=L_{\nu }^p\left(F\right)$, $1\le p<+\infty$. Going back to the moment conditions in Equation (2), due to Haviland’s theorem [11], for the existence of such a positive linear functional $T$ satisfying the moment conditions, it is sufficient (and necessary) that the linear form

$$T_0:\mathcal{P}\to \mathbb{R}, {\mathrm{T}}_0\left({\displaystyle \sum }_{\mathrm{j}\in {\mathrm{J}}_0}{\mathsf{\alpha }}_{\mathrm{j}}{\mathsf{\phi }}_{\mathrm{j}}\right)≔{\displaystyle \sum }_{\mathrm{j}\in {\mathrm{J}}_0}{\mathsf{\alpha }}_{\mathrm{j}}{\mathrm{y}}_{\mathrm{j}},$$

(3)

defined on the subspace $\mathcal{P}$ of polynomial functions, satisfies the condition

$$\mathrm{p}\in \mathcal{P}, \mathrm{p}\left(\mathrm{t}\right)\ge 0 \forall \mathrm{t}\in F\Rightarrow T_0\left(\mathrm{p}\right)\ge 0.$$

(4)

If the implication (4) holds true, Haviland’s theorem ensures the existence of a positive regular Borel measure $\mu$ on $F$, such that

$${\mathrm{y}}_{\mathrm{j}}=T_0\left({\mathsf{\phi }}_{\mathrm{j}}\right)={\int }_F{\mathrm{t}}^{\mathrm{j}}\mathrm{d}\mathsf{\mu }, \mathrm{j}\in {\mathbb{N}}^{\mathrm{n}}.$$

(5)

If Equation (5) holds, we say that ${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$ is a moment sequence (or a sequence of moments) on $F.$ In the case of a Markov moment problem, more powerful extension results for linear functionals can be applied. Aside from positivity, the extension $T$ of $T_0$ is dominated by a given continuous sublinear functional $P$ (or by a continuous convex functional) on $X$. Usually, this leads not only to the continuity of $T,$ but also to evaluating (or even determining exactly) its norm, in terms of the norm of the continuous sublinear functional $P$. Alternately, the sandwich condition appearing in the Markov moment problem is formulated as:

$$T_2\left(\mathsf{\psi }\right)\le T\left(\mathsf{\psi }\right)\le T_3\left(\mathsf{\psi }\right) \forall \mathsf{\psi }\in {\mathrm{X}}_{+}.$$

(6)

Here $T_2, T_3$ are given bounded linear functionals defined on $X$.

If the $y_j$ are elements of a Banach lattice, we have a classical vector valued moment problem or a classical vector valued Markov moment problem. Then $Y$ is assumed to be an order complete Banach lattice, $P:X=L_{\nu }^p\left(F\right)\to Y (1\le p<+\infty )$ a continuous sublinear (or only convex) operator, $T_0:\mathcal{P}\left(F\right)\to \mathrm{Y}, T_1:X_1\to Y, T:X\to Y$ are linear operators. In this operator setting, observe that $\mathcal{P}\left(F\right)$ is a majorizing subspace of $X_1$ and, according to (4), $T_0$ is a positive linear operator. Application of Kantorovich theorem [12] leads to the existence of a positive linear extension $T_1:X_1\to Y$ of $T_0.$ In this framework, we have linear operators in Equation (6). The general idea is to prove that if $X=L_{\nu }^1\left(F\right), \nu$ is moment determinate, $T_2,T_3$ are bounded linear operators applying $X$ into $Y$, and, using the above notations,

$$T_2\left(p\right)\le T\left(p\right)\le T_3\left(p\right) \forall p\in {\mathcal{P}}_{++}\left(F\right),$$

then the extension $T_1:X_1\to Y$ verifies

$$T_2\left(\psi \right)\le T_1\left(\psi \right)\le T_3\left(\psi \right) \forall \psi \in {\left(C_c\left(F\right)\right)}_{+}.$$

(7)

This can be proved by means of approximation of $\psi$ with dominating polynomials from ${\mathcal{P}}_{++}\left(F\right).$ If $F={\mathbb{R}}^n$, $\nu ={\nu }_1\times \cdots \times {\nu }_n, {\nu }_j$ is moment determinate on $\mathbb{R}, j=1,\dots ,n,$ then applying (7), it is easy to deduce the continuity of $T_1$ on $C_c\left(F\right)$. Thus $T_1$ admits a unique continuous extension $T:X\to Y,$ which satisfies the moment interpolation conditions $T\left({\mathsf{\phi }}_{\mathrm{j}}\right)=T_1\left({\mathsf{\phi }}_{\mathrm{j}}\right)={\mathrm{y}}_{\mathrm{j}}, \mathrm{j}\in {\mathbb{N}}^{\mathrm{n}}$ and the sandwich condition $T_2\left(f\right)\le T\left(f\right)\le T_3\left(f\right)$ for all $f\in X_{+},$ since ${\left(C_c\left(F\right)\right)}_{+}$ is dense in $X_{+}.$ Thus, the difficulty appearing because of the unknown form of nonnegative polynomials on ${\mathbb{R}}^n$ is solved for the full Markov moment problems, via polynomial approximation with elements from ${\mathcal{P}}_{++}\left({\mathbb{R}}^n\right)$. This is the second aim of the present work. Recall that these polynomials are finite sums of tensor products defined by (1), where each $p_j$ is the sum of two squares of polynomials on $\mathbb{R},$ hence the sums of polynomials (1) are sums of squares.

The moment problems discussed up to now are full moment problems since they involve the moments of all orders. If we require the interpolation conditions $T\left({\mathsf{\phi }}_{\mathrm{j}}\right)=y_j$ only for a finite number of moments (usually one writes $j=\left(j_1,\dots ,j_n\right), j_k\le d$ for all $k\in \left\{1,\dots ,n\right\}$ for some fixed positive integer $d$), then we have a truncated (reduced) moment problem. Various aspects of the moment full or reduced moment problem and related problems are studied in references [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. In [29], among other results and associated examples, new applications of various extension theorems for linear operators with two constraints are proved. Moreover, in [6,29], one proves that any positive linear operator acting between two ordered Banach spaces is continuous. In particular, this result is valid for positive linear operators acting between Banach lattices. For approximation and density theorems not necessarily related to the moment problem see the article [30]. The papers [31,32,33,34,35,36,37], refer to the Markov moment problem. Other Markov moment problems are discussed in references [4,16]. We start with the case of compact subsets for which the form of positive polynomials is expressible in terms of special polynomials defining those compact sets. As a new element, we solve Markov vector valued moment problem on such subsets. Then we continue with full Markov moment problems on unbounded subsets. Both types of such problems are solved by polynomial approximation. In most of the cases, this method leads to characterizations of the existence and uniqueness of the solution for each involved problem, in terms of quadratic forms or other simple ways of writing. The rest of this study is organized as follows. In Section 2, the useful methods applied along this paper are briefly reviewed. Section 3 contains old and recent results on the subject, accompanied by a few consequences. Section 4 concludes the paper.

2. Methods

Here are the basic methods applied along the proofs of the results of this work.

(I)

Polynomial approximation on some compact subsets $K\subset {\mathbb{R}}^n$, in the spaces $C\left(K\right)$ and $L_{\nu }^1\left(F\right)$, where $\nu$ is a positive regular Borel measure on $K,$ and related moment problems cf. [14,15,16,17,18,19,35]). We observe that any such measure is moment determinate, due to the Weierstrass uniform approximation theorem with polynomials. (cf. [14,15,16,17,18,19,35]).

(II)

Polynomial approximation of any nonnegative function $\psi \in L_{\nu }^1\left(F\right)$ with dominating polynomials. When $F={\mathbb{R}}^n$, (respectively $F={\mathbb{R}}_{+}^n)$, and $\mathit{\nu }$ is the product of $\mathit{n}$ moment determinate measures on $\mathbb{R}$ (respectively on ${\mathbb{R}}_{+}$), such approximation holds true with sums of tensor products defined by Equation (1), where each factor ${\mathit{p}}_{\mathit{j}}$ is nonnegative on $\mathbb{R},$ (respectively on ${\mathbb{R}}_{+}$). Applications to the Markov moment problem on unbounded subsets $j=1,\dots ,n.$ (cf. [34,35,37]).

(III)

Hahn-Banach extension theorems for linear operators with one or two constraints (sandwich condition) on the linear extension $\mathit{T}$ (cf. [10,12,16]).

(IV)

Properties and examples of positive linear operators acting between Banach lattices (cf. [6,29]).

(V)

Applying a sufficient condition for determinacy to a class of measures on ${\mathbb{R}}_{+}$ (cf. [24]).

(VI)

Applying elements of the theory of self-adjoint operators and a result on a space of such operators, proved in [5].

3. Results

3.1. Markov Moment Problems on Compact Subsets

The following results are reviewed in the chronological order of the dates of their publication. A few completions are added, stated, and eventually proved as corollaries. We start with a short presentation of some of the results published firstly in [14]. Let $K$ be a compact subset of ${\mathbb{R}}^n,$ with a non-empty interior. The open set $D={\mathbb{R}}^n\backslash K$ can be written as a union of sets of the form ${\cup }_{i\in I}{p}_i^{-1}\left(-\infty ,0\right)$, hence $K={\cap }_{i\in I}{p}_i^{-1}\left({\mathbb{R}}_{+}\right),$ where $p_i, i\in I$ are polynomials of degrees smaller or equal to two. One denotes by $E\left(K\right)$ the vector space generated by the polynomials of degree at most one and the polynomials $p_i, i\in I.$ The space $E\left(K\right)$ is clearly a vector subspace of the space ${\mathbb{R}}_2\left[t_1,\dots ,t_n\right]$ of all polynomials of degrees at most two. We denote by $E_{+}\left(K\right)$ the convex cone of all polynomials in $E\left(K\right)$, which takes non-negative values at all points of $K$, and one denotes by $G\left(K\right)$ the set of those $p\in E_{+}\left(K\right)$ that generates an extreme ray of $E_{+}\left(K\right).$ An important subset is

$${\mathit{G}}_{\mathbf{1}}\mathbf{\left(}\mathit{K}\mathbf{\right)}\mathbf{=}\mathbf{\left\{}\mathit{p}\mathbf{\in }\mathit{G}\mathbf{\left(}\mathit{K}\mathbf{\right)}\mathbf{;}{\mathbf{\Vert }\mathit{p}\mathbf{\Vert }}_{\mathit{K}}\mathbf{=}\mathbf{1}\mathbf{\right\}}.$$

If $K$ is convex, then we can take as $E\left(K\right)$ the space ${\mathbb{R}}_1\left[t_1,\dots ,t_n\right]$ of all polynomials of degree at most one. With $n=1$ and $K=\left[0,1\right]$, it is easy to see that $G_1\left(K\right)$ consists in only two elements: the polynomials $t$ and $1-t.$ From this, we infer that the set of polynomials $t^k{\left(1-t\right)}^l, k,l\in \mathbb{N},$ which appear naturally in the classical Hausdorff moment problem, should be replaced, in the general case, by the set $\Delta \left(K\right)$ of polynomials, which are finite products of elements of $G_1\left(K\right).$ Since $\Delta \left(K\right)$ has only a multiplicative structure (the sum of two elements of this set is not, in general, an element of $\Delta \left(K\right)$), one introduces the convex cone of all linear combinations with nonnegative coefficients of elements of $\Delta \left(K\right).$ The next result gives a necessary and sufficient condition for the existence of a solution to the moment problem. This condition is formulated only in terms of the moments $y_j, j\in {\mathbb{N}}^n$ and the special polynomials that are elements of $\Delta \left(K\right).$ Therefore, we say that the next theorem solves the moment problem. From the point of view of the next theorem, it represents the multidimensional case of the Hausdorff moment problem (the moment problem on $K=\left[0,1\right]$). On the other hand, the next theorem works for non-convex compact subsets having non-empty interiors as well. We say that a sequence of real numbers ${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$ is a moment sequence on the closed subset $F\subseteq {\mathbb{R}}^n$ if there exists a positive regular Borel measure $\mu$ on $F$ such that $y_j={\int }_F{t}^{j}d\mu , j\in {\mathbb{N}}^n.$

Theorem 1.

(See [14]). Let $\mathit{K}$ be a compact subset of ${\mathbb{R}}^{\mathit{n}},$ with a non-empty interior. A necessary and sufficient condition for a sequence ${\left({\mathit{y}}_{\mathit{j}}\right)}_{\mathit{j}\in {\mathbb{N}}^{\mathit{n}}}$ being a moment sequence on $\mathit{K}$ is that the linear form $\mathit{T}$ defined on $\mathbb{R}[{\mathit{t}}_{\mathit{1}},\dots ,{\mathit{t}}_{\mathit{n}}]$ by $\mathit{T}\left({\mathit{t}}^{\mathit{j}}\right)={\mathit{y}}_{\mathit{j}}, \mathit{j}\in {\mathbb{N}}^{\mathit{n}}$ satisfies the condition $\mathit{T}\left(\mathit{p}\right)\ge 0$ for all polynomial $\mathit{p}\in \Delta \left(\mathit{K}\right)$.

The next result gives the expression of any polynomial $p$ that is positive at each point of $K$, by means of some polynomials that are elements of $\Delta \left(K\right).$ Since any such polynomial $p$ is a linear combination with nonnegative coefficients of elements of $\Delta \left(K\right).$ (see below), the next result is called the decomposition theorem.

Theorem 2.

(See [14]). Each polynomial that has positive values at all points of a compact subset $\mathit{K}$ with a non-empty interior in ${\mathbb{R}}^{\mathit{n}}$ is a linear combination with nonnegative coefficients of elements of $\Delta \left(\mathit{K}\right).$

Here is a vector valued version of a solution of a Markov moment problem similar to the solution for the Hausdorff moment problem. In this case, we denote

$${\phi }_j\left(t\right)=t^j, t\in \left[0,1\right], j\in \mathbb{N}.$$

Next, we pass to the vector valued Markov moment problem for compact subsets $K\subset {\mathbb{R}}^n,$ with nonempty interior.

Theorem 3.

Let$Y$be an arbitrary order complete Banach lattice,${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$a sequence in$Y, T_1, T_2$two linear operators from$X≔C\left(K\right)$into$Y,$such that$\mathbf{0}\le T_1\le T_2$on$X_{+}.$The following statements are equivalent:

(a)

There exists a unique bounded linear operator$T$applying$X$into$Y$such that

$$T\left({\phi }_j\right)=y_j \forall j\in {\mathbb{N}}^{\mathit{n}}, T_1\le T\le T_2 \mathit{on} X_{+}, \Vert T_1\Vert \le \Vert T\Vert \le \Vert T_2\Vert .$$

(b)

The following inequalities hold:

$$T_1\left(p\right)\le T_0\left(p\right)\le T_2\left(p\right) \forall p\in \Delta \left(K\right),$$

(8)

where$T_0:\mathcal{P}\left(K\right)\to Y$is defined by (3).

Proof. 

Sine all the polynomials in $\Delta \left(K\right)$ are elements of $X_{+},$ the implication (a)$\Rightarrow$(b) is obvious. Here $T_0:\mathcal{P}\left(K\right)\to Y$ is the unique linear operator from $\mathcal{P}\left(K\right)$ into $Y$ satisfying the moment conditions $T_0\left({\phi }_j\right)=y_j$ for all $j\in {\mathbb{N}}^n.$ To prove the implication (b)$\Rightarrow$(a), we need Theorem 2. Let us denote by ${\mathcal{P}}_{++}\left(K\right)$ the set of all polynomials $q$ which take only positive values at each point of $K.$ If $p\in {\mathcal{P}}_{+}\left(K\right),$ then clearly $q=p+\epsilon 1\in {\mathcal{P}}_{++}\left(K\right)$ for all $\epsilon >0$ Conversely, if $q\in {\mathcal{P}}_{++}\left(K\right),$ then $q=(q-\underset{t\in K}{\min }q\left(t\right))+\underset{t\in K}{\min }q\left(t\right))=p+\epsilon ,$ where

$$\epsilon ≔\underset{t\in \left[0,1\right]}{\min }q\left(t\right)>0, p≔q-\epsilon \mathbf{1}\in {\mathcal{P}}_{+}\left(K\right).$$

On the other hand, according to Theorem 2, any polynomial $q$ from ${\mathcal{P}}_{++}\left(K\right)$ is a linear combination with nonnegative coefficients of polynomials from $\Delta \left(K\right).$ Thus, inequalities (8) lead to

$$T_1\left(q\right)\le T_0\left(q\right)\le T_2\left(q\right) \forall q\in {\mathcal{P}}_{++}\left(K\right).$$

The first conclusion is:

$$T_1\left(p\right)+\epsilon T_1\left(\mathbf{1}\right)\le T_0\left(p\right)+\epsilon T_0\left(\mathbf{1}\right)=T_0\left(p\right)+\epsilon y_0\le T_2\left(p\right)+\epsilon T_2\left(\mathbf{1}\right), \forall p\in {\mathcal{P}}_{+}\left(K\right), \forall \epsilon >0.$$

Passing to the limit as $\epsilon \downarrow 0,$ we find that:

$$\mathbf{0}\le T_1\left(\mathrm{p}\right)\le {\mathrm{T}}_0\left(\mathrm{p}\right)\le {\mathrm{T}}_2\left(\mathrm{p}\right) \forall \mathrm{p}\in {\mathcal{P}}_{+}\left(K\right)≔\left\{\mathrm{p}\in \mathcal{P};\mathrm{p}\left(\mathrm{t}\right)\ge 0 \forall \mathrm{t}\in \mathrm{K}\right\}.$$

(9)

Hence $T_0$ is a positive linear operator from $\mathcal{P}\left(K\right)$ into $Y.$ According to Kantorovich extension theorem [12], there exists a positive linear extension $T:C\left(K\right)\to Y$ of $T_0$, because $\mathcal{P}\left(K\right)$ is a majorizing subspace in $C\left(K\right).$ The operators $T_1,T, T_2$ are continuous, since they are positive, acting between Banach lattices ([6] or [29]). If $f\in X_{+},$ there exists a sequence ${\left(p_m\right)}_{m\in \mathbb{N}}, p_m\in {\mathcal{P}}_{+}\left(K\right)$ for all $m, p_m\downarrow f$ in $C\left(K\right).$ Using (9) and the continuity of the involved operators, this leads to:

$$T_1\left(f\right)=\underset{m}{\lim }{T}_1\left(p_m\right)\le \underset{m}{\lim }{T}_0\left(p_m\right)=\underset{m}{\lim }T\left(p_m\right)=T\left(f\right)\le \underset{m}{\lim }{T}_2\left(p_m\right)=T_2\left(f\right).$$

Thus, $T_1\le T\le T_2$ on $X_{+}.$ To prove the last inequalities in the statement, observe that

$$\pm T_1\left(g\right)=T_1\left(\pm g\right)\le T_1\left(\left|g\right|\right)\Rightarrow \left|T_1\left(g\right)\right|\le T\left(\left|g\right|\right)\Rightarrow$$

$$\Vert T_1\left(g\right)\Vert \le \Vert T\left(\left|g\right|\right)\Vert \le \Vert T\Vert ·\Vert g\Vert \forall g\in C\left(K\right)\Rightarrow \Vert T_1\Vert \le \Vert T\Vert .$$

The inequality $\parallel T\parallel \le \parallel T_2\parallel$ follows by means of the same type of reasons. Finally, the operator $T$ clearly verifies the interpolation moment conditions stated at point (a), since

$$T\left({\phi }_j\right)=T_0\left({\phi }_j\right)≔y_j, j\in {\mathbb{N}}^{\mathit{n}}.$$

This concludes the proof. □

Now the Hausdorff Markov vector valued moment problem follows consequently.

Theorem 4.

Let$Y$be an arbitrary order complete Banach lattice,${\left(y_j\right)}_{j\in \mathbb{N}}$a sequence in$Y, T_1, T_2$two linear operators from$X≔C\left(\left[0,1\right]\right)$into$Y,$such that$\mathbf{0}\le T_1\le T_2$on$X_{+}$. The following statements are equivalent:

(a)

There exists a unique bounded linear operator$T$applying$X≔C\left(\left[0,1\right]\right)$into$Y$such that

$$T\left({\phi }_j\right)=y_j \forall j\in \mathbb{N}, T_1\le T\le T_2 on X_{+}, \parallel T_1\parallel \le \parallel T\parallel \le \parallel T_2\parallel .$$

(b)

For any $k,l\in \mathbb{N},$ the following inequalities hold:

$${\displaystyle \sum }_{j=0}^l{\left(-1\right)}^j\left(\begin{array}{c}l\\ j\end{array}\right)T_1\left({\phi }_{k+j}\right)\le {\displaystyle \sum }_{j=0}^l{\left(-1\right)}^j\left(\begin{array}{c}l\\ j\end{array}\right)y_{k+j} \le {\displaystyle \sum }_{j=0}^l{\left(-1\right)}^j\left(\begin{array}{c}l\\ j\end{array}\right)T_2\left({\phi }_{k+j}\right) .$$

Proof. 

We start by observing that ${\phi }_k\left(t\right){\phi }_j\left(t\right)=t^k{t}^j=t^{k+j}={\phi }_{k+j}\left(t\right)$ for all $t,$ hence ${\phi }_k{\phi }_j={\phi }_{k+j}$ for all $k,j\in \mathbb{N}$. Thus, point (b) says that

$$T_1\left({\phi }_k{\left(1-{\phi }_1\right)}^l\right)\le T_0\left({\phi }_k{\left(1-{\phi }_1\right)}^l\right)\le T_2\left({\phi }_k{\left(1-{\phi }_1\right)}^l\right), k,l\in \mathbb{N}.$$

This assertion is equivalent to that form (b) of Theorem 3, since we have

$$\Delta \left(\left[0,1\right]\right)=\left\{{\phi }_k{\left(1-{\phi }_1\right)}^l;k,l\in \mathbb{N}\right\}.$$

The conclusion follows. □

The next result is an application of theorem 4 to the case when the order complete Banach lattice is a space of self-adjoint operators acting on an arbitrary real or complex Hilbert space $H, \dim \left(H\right)\ge 2.$ We recall that a linear operator $A:D\left(A\right)\subseteq H\to H,$ (where $D\left(A\right)$ is a vector subspace of $H$), is called self-adjoint if it is bounded (continuous) and symmetric, that is $⟨A\left(h_1\right),h_2⟩=⟨h_1,A\left(h_2\right)⟩$ for all $h_1,h_2\in D\left(A\right).$ According to this definition, any self-adjoint operator is symmetric. When $D\left(A\right)=H,$ the converse is true, since any symmetric operator from $H$ into itself is bounded, according to the closed graph theorem. A self-adjoint operator $A$ acting on $H$ is called positive if $⟨A\left(h\right),h⟩ \ge 0$ for all $h\in H.$ The real vector space of all self-adjoint operators acting on $H$ will be denoted by $\mathcal{A}\left(H\right).$ If $A\in \mathcal{A}\left(H\right)$, the following equalities for the operatorial norm of $A$ holds:

$$\Vert A\Vert ≔\underset{\Vert h\Vert \le 1}{sup}\Vert A\left(h\right)\Vert =\underset{\Vert h\Vert \le 1}{sup}\left|⟨A\left(h\right),h⟩\right|.$$

For any $A\in \mathcal{A}\left(H\right)$, we define

$${\mathrm{Y}}_1\left(\mathrm{A}\right)≔\left\{\mathrm{V}\in \mathcal{A}\left(\mathrm{H}\right);\mathrm{AV}=\mathrm{VA}\right\}, \mathrm{Y}\left(\mathrm{A}\right)≔\left\{\mathrm{W}\in {\mathrm{Y}}_1\left(\mathrm{A}\right);\mathrm{WU}=\mathrm{UW} \forall \mathrm{U}\in {\mathrm{Y}}_1\left(\mathrm{A}\right)\right\}.$$

(10)

In [5], it was proved that $Y\left(A\right)$ is an order complete Banach lattice (which is also a commutative real Banach algebra). In the case when $H={\mathbb{R}}^n, n\ge 2,$ there exists a natural isomorphism between the ordered Banach space $\mathcal{A}\left(H\right)$ and the real vector space of all symmetric $n\times n$ matrices with real entries. If $A,B$ are two such matrices, then, according to the above definitions, $A\le B$ if and only if $B-A$ is positive semi-definite. For an arbitrary Hilbert space $H,$ with $dimH\ge 2,$ the ordered Banach space $\mathcal{A}\left(H\right)$ is not a lattice. If $\sigma \left(A\right)$ is the spectrum of $A\in \mathcal{A}\left(H\right)$ and $f\in C\left(\sigma \left(A\right)\right),$ we denote by $f\left(A\right)$ the element obtained by means of functional calculus. Namely, $f\left(A\right)={\int }_{\sigma \left(A\right)}f\left(t\right)d{E}_A,$ where $d{E}_A$ is the spectral measure attached to $A.$

Corollary 1.

Let$H$be a Hilbert space,$\dim \left(H\right)\ge 2, A\in \mathcal{A}\left(H\right)$with the spectrum$\sigma \left(A\right)=\left[0,1\right],$$X:C\left(\left[0,1\right]\right),$let$m$be a positive integer and$Y\left(A\right)$be defined by (10). Let${\left(B_j\right)}_{j\in \mathbb{N}}$be a sequence of elements in$Y\left(A\right).$The following statements are equivalent:

(a)

There exists a unique bounded linear operator$T$applying$X≔C\left(\left[0,1\right]\right)$into$Y\left(A\right)$such that

$$T\left({\phi }_j\right)=B_j \forall j\in \mathbb{N}, A^{m}f\left(A\right)\le T\left(f\right)\le f\left(A\right) \forall f\in X_{+}, \Vert A^m\Vert \le \Vert T\Vert \le 1.$$

(b)

For any$k,l\in \mathbb{N},$the following inequalities hold:

$${\displaystyle \sum }_{j=0}^l{\left(-1\right)}^j\left(\begin{array}{c}l\\ j\end{array}\right)A^{m+k+j}\le {\displaystyle \sum }_{j=0}^l{\left(-1\right)}^j\left(\begin{array}{c}l\\ j\end{array}\right)B_{k+j} \le {\displaystyle \sum }_{j=0}^l{\left(-1\right)}^j\left(\begin{array}{c}l\\ j\end{array}\right)A^{k+j}.$$

In the sequel, moment problems on semi-algebraic compact subsets are reviewed. If $y={\left(y_j\right)}_{j\in {\mathbb{N}}^n}$ $n\ge 2$ is a sequence of real numbers, one denotes by $T_y$ the linear functional defined on $\mathcal{P}=\mathbb{R}\left[t_1,\dots ,t_n\right]$ by

$$T_y\left({\displaystyle \sum }_{j\in J_0}{\alpha }_j{t}^j\right)={\displaystyle \sum }_{j\in J_0}{\alpha }_j{y}_j,$$

where $J_0\subset {\mathbb{N}}^n$ is a finite subset and ${\alpha }_j$ are arbitrary real coefficients. If $\left\{f_1,\dots ,f_k\right\}$ is a finite subset of $\mathbb{R}\left[t_1,\dots ,t_n\right]$, then the closed subset given by

$$\mathrm{K}=\left\{\mathrm{t}\in {ℝ}^{\mathrm{n}};{\mathrm{f}}_1\left(\mathrm{t}\right)\ge 0,\dots ,{\mathrm{f}}_{\mathrm{k}}\left(\mathrm{t}\right)\ge 0\right\}$$

(11)

is called a semi-algebraic set.

Theorem 5.

(See [4,15]). If $K$ is a compact semi-algebraic set, as defined above, then there is a positive Borel measure $\mu$ supported on $K$, such that

$${{\displaystyle \int }}_K{t}^{j}d\mu =y_j, \forall j\in {\mathbb{N}}^n,$$

if, and only if:

$$T_y\left(f_1^{e_1}\cdots f_k^{e_k}{p}^2\right)\ge 0, \forall p\in \mathbb{R}\left[t_1,\dots ,t_n\right], \forall e_1,\dots ,e_k\in \left\{0,1\right\}.$$

Corollary 2.

(See [15]). With the above notations, if $p\in \mathbb{R}\left[t_1,\dots ,t_n\right]$ is such that $p\left(t\right)>0$ for all $t$ in the semi-algebraic compact $K$ defined by Equation (11), then $p$ is a finite sum of special polynomials of the form

$$f_1^{e_1}\cdots f_k^{e_k}{q}^2\ge 0,$$

for some $q\in \mathbb{R}\left[t_1,\dots ,t_n\right]$ and $e_1,\dots ,e_k\in \left\{0,1\right\}.$

Corollary 1 is named Schmüdgen’s Positivstellensatz. There also exists Putinar’s Positivstellensatz. These are representations of positive polynomials on basic closed semi-algebraic sets, in terms of sums of squares and polynomials defining the semi-algebraic set under attention. For other results on this subject, as well as for Markov moment problems not reviewed in the present work see [3,4,17,18,19,20,33]. In [19], moment problems on unbounded subsets are also discussed.

Lemma 1

(See [36]). Let $\psi :{\mathbb{R}}_{+}=\left[0,\infty \right)\to {\mathbb{R}}_{+}$ be a continuous function, such that $\underset{t\to \infty }{\mathit{lim}}\psi \left(t\right)$ exists in ${\mathbb{R}}_{+}.$ Then, there is a decreasing sequence ${\left(h_l\right)}_l$ in $Span\left\{e_k;k\in \mathbb{N}\right\}$, where the functions $e_k;k\in \mathbb{N}$ are defined as follows:

$$e_k\left(t\right)=exp\left(-kt\right), k\in \mathbb{N}, t\in \left[0,\infty \right),$$

such that $h_l\left(t\right)\ge \psi \left(t\right),$ $t\ge 0, l\in \mathbb{N}=\left\{0,1,2,\dots \right\},$ $lim{h}_l=\psi$ uniformly on $\left[0,\infty \right]$. There exists a sequence of polynomial functions ${\left({\tilde{p}}_l\right)}_{l\in \mathbb{N}}$, ${\tilde{p}}_l\ge h_l\ge \psi , lim {\tilde{p}}_l=\psi$, uniformly on compact subsets of $\left[0,\infty \right)$. In particular, such polynomial approximation holds for any continuous compactly supported function $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$.

If $K\subset {\mathbb{R}}_{+}$ is an arbitrary compact subset, then for any function $\phi :K\to {\mathbb{R}}_{+}$, one denotes by ${\phi }_0:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ the extension of $\phi$, which satisfies ${\phi }_0\left(t\right)=0$ for all $t\in {\mathbb{R}}_{+}\backslash K$.

Lemma 2

(See [36] and the references therein). If $K\subset {\mathbb{R}}_{+}$ is a compact subset, and $\phi :K\to {\mathbb{R}}_{+}$ a continuous function, then there exists a sequence ${\left({\tilde{p}}_l\right)}_{l\in \mathbb{N}}$ of polynomial functions, such that ${\tilde{p}}_l\ge {\phi }_0$ on ${\mathbb{R}}_{+},{{\tilde{p}}_l|}_K\to \phi ,l\to \infty$, uniformly on $K.$

Corollary 3.

Let$K\subset {\mathbb{R}}_{+}$be a compact subset,$\phi :K\to {\mathbb{R}}_{+}$a continuous function, $Y$a Banach lattice. Assume that$T_1,T,T_2$are positive linear operators from$X≔C\left(K\right)$into$Y.$The following statements are equivalent:

(a)

The following sandwich conditions hold:$T_1\le T\le T_2$on$X_{+}, \Vert T_1\Vert \le \Vert T\Vert \le \Vert T_2\Vert .$

(b)

For any finite subset$J_0\subset \mathbb{N}$, and any$\left\{{\lambda }_j;j\in J_0\right\}\subset \mathbb{R},$the following inequalities hold:

$${\displaystyle \sum }_{i,j\in J_0}{\lambda }_i{\lambda }_j{T}_1\left({\phi }_{i+j+k}\right)\le {\displaystyle \sum }_{i,j\in J_0}{\lambda }_i{\lambda }_{j}T\left({\phi }_{i+j+k}\right)\le {\displaystyle \sum }_{i,j\in J_0}{\lambda }_i{\lambda }_j{T}_2\left({\phi }_{i+j+k}\right), k\in \left\{0,1\right\}.$$

Proof. 

The implication $\left(\mathrm{a}\right)\Rightarrow \left(\mathrm{b}\right)$ is obvious since any polynomial

$${\displaystyle \sum }_{i,j\in J_0}{\lambda }_i{\lambda }_j{\phi }_{i+j+k}, k\in \left\{0,1\right\},$$

(12)

is nonnegative on $K\subset {\mathbb{R}}_{+}$. For the converse, we know form Lemma 2 that there exists a sequence ${\left({\tilde{p}}_l\right)}_{l\in \mathbb{N}}$ of nonnegative polynomials on ${\mathbb{R}}_{+}, {\tilde{p}}_l{|}_K\to \phi , l\to \infty$, uniformly on $K$. Condition (b) implies $T_1\left({\tilde{p}}_l\right)\le T\left({\tilde{p}}_l\right)\le T_2\left({\tilde{p}}_l\right)$ for all $l\in \mathbb{N},$ because each nonnegative polynomial on ${\mathbb{R}}_{+}$ has the analytic form given in (12). Now the first part of assertion (a) follows from the continuity of the three involved positive operators $T_1,T,T_2$. Namely, we have:

$$T_1\left(\phi \right)=\underset{l}{\lim }{T}_1\left({\tilde{p}}_l\right)\le \underset{l}{\lim }T\left({\tilde{p}}_l\right)=T\left(\phi \right)\le \underset{l}{\lim }{T}_2\left({\tilde{p}}_l\right)=T_2\left(\phi \right).$$

The inequalities $\Vert T_1\Vert \le \Vert T\Vert \le \Vert T_2\Vert$ have been established in the proof of Theorem 3. This concludes the proof. □

Corollary 4.

Let$H$be a Hilbert space, and$A\in \mathcal{A}\left(H\right)$a self-adjoint operator acting on$H,$with the spectrum$\sigma \left(A\right)\subseteq \left[0,1\right].$Let$Y=Y\left(A\right)$be the Banach lattice defined by (10) and let denote$X≔C\left(\sigma \left(A\right)\right)$. The following statements are equivalent.

(a)

We have

$$\left(I-A\right)\phi \left(A\right)\le e^{-A}\phi \left(A\right)\le \phi \left(A\right) \forall \phi \in X_{+}, \Vert I-A\Vert \le \Vert e^{-A}\Vert \le 1.$$

(b)

For any finite subset$J_0\subset \mathbb{N}$, and any$\left\{{\lambda }_j;j\in J_0\right\}\subset \mathbb{R},$the following inequalities hold:

$$\left(I-A\right){\displaystyle \sum }_{i,j\in J_0}{\lambda }_i{\lambda }_j{A}^{i+j+k}\le e^{-A}{\displaystyle \sum }_{i,j\in J_0}{\lambda }_i{\lambda }_j{A}^{i+j+k}\le {\displaystyle \sum }_{i,j\in J_0}{\lambda }_i{\lambda }_j{A}^{i+j+k}, k\in \left\{0,1\right\}.$$

Proof. 

We apply Corollary 3 to

$$T_1\left(\phi \right)≔{{\displaystyle \int }}_{\sigma \left(A\right)}\left(1-t\right)\phi \left(t\right)d{E}_A=\left(I-A\right)\phi \left(A\right), T\left(\phi \right)≔{{\displaystyle \int }}_{\sigma \left(A\right)}{e}^{-t}\phi \left(t\right)d{E}_A=e^{-A}\phi \left(A\right),$$

$$T_2\left(\phi \right)≔{{\displaystyle \int }}_{\sigma \left(A\right)}\phi \left(t\right)d{E}_A=\phi \left(A\right), \phi \in C\left(\sigma \left(A\right)\right).$$

One also uses the elementary inequalities

$$0\le 1-t\le e^{-t}\le 1 \mathrm{for} \mathrm{all} t\in \sigma \left(A\right)\subseteq \left[0,1\right].$$

It follows that $\mathbf{0}\le I-A\le e^{-A}\le I,$ hence $\Vert I-A\Vert \le \Vert e^{-A}\Vert \le 1,$ since the norm is monotone increasing on the positive cone of $Y\left(A\right).$ This concludes the proof. □

Example 1.

We consider the real Hilbert space$H≔L^2\left[0,1\right], A\in \mathcal{A}\left(H\right)$the operator defined by$A\left(h\right)\left(t\right)≔th\left(t\right), t\in \left[0,1\right], h\in H$. Then$A$is a positive (linear) symmetric operator,

$$\sigma \left(A\right)=\left[0,1\right], \sigma \left(I-A\right)=\left\{1-t;t\in \sigma \left(A\right)\right\}=\left[0,1\right]\Rightarrow \Vert I-A\Vert =1,$$

$$\sigma \left(e^{-A}\right)=\left[e^{-1},1\right]\Rightarrow \Vert e^{-A}\Vert =1,$$

$$A^m\left(h\right)\left(t\right)=t^{m}h\left(t\right), t\in \left[0,1\right], h\in L^2\left[0,1\right], m\in \mathbb{N}.$$

We go on with a general Hahn-Banach type result, namely Mazur-Orlicz theorem, followed by an application.

Theorem 6.

(See [10,16,36]). Let $X$ be a preordered vector space, $Y$ an order complete vector space, $P:X\to Y$ a sublinear operator, and ${\left\{x_j\right\}}_{j\in J}\subset X, {\left\{y_j\right\}}_{j\in J}\subset Y$ given families. The following statements are equivalent:

(a)

There exists a linear positive operator$T:X\to Y$, such that

$$T\left(x_j\right)\ge y_j, j\in J, T\left(x\right)\le P\left(x\right), x\in X.$$

(b)

For any finite subset$J_0\subseteq J, and any \left\{{\lambda }_j;j\in J_0\right\}\subseteq {\mathbb{R}}_{+}$, we have

$${\displaystyle \sum }_{j\in J_0}{\lambda }_j{x}_j\le x\in X\Rightarrow {\displaystyle \sum }_{j\in J_0}{\lambda }_j{y}_j\le P\left(x\right).$$

Let $H$ be a Hilbert space, $A\in \mathcal{A}\left(H\right)$ a positive self-adjoint operator acting on $H,\sigma \left(A\right)$ the spectrum of $A, E_A$ the spectral measure attached to $A.$ Let $Y=Y\left(A\right)$ be the order complete Banach lattice defined by (10), and ${\left(B_j\right)}_{j\in \mathbb{N}}$ a sequence of elements in $Y\left(A\right).$ As usual, we denote ${\phi }_j\left(t\right)≔t^j, t\in \sigma \left(A\right), j\in \mathbb{N}.$

Theorem 7.

(See [10] and the references therein). The following statements are equivalent.

(a)

There exists a positive linear operator$T$from$C\left(\sigma \left(A\right)\right)$into$Y\left(A\right)$, such that

$$T\left({\phi }_j\right)\ge B_{j } \forall j\in \mathbb{N}, T\left(f\right)\le {{\displaystyle \int }}_{\sigma \left(A\right)}\left|f\left(t\right)\right|d_{E_A} \forall f\in C\left(\sigma \left(A\right)\right), \Vert T\Vert \le 1.$$

(b)

$B_j\le A^j$for all$j\in \mathbb{N}.$

Proof. 

Since the self-adjoint operator $A$ is positive, its spectrum $\sigma \left(A\right)$ is contained in ${\mathbb{R}}_{+}.$ The implication $\left(a\right)\Rightarrow \left(b\right)$ is almost obvious. Namely, if (a) holds, then:

$$B_j\le T\left({\phi }_j\right)\le {{\displaystyle \int }}_{\sigma \left(A\right)}\left|t^j\right|d_{E_A}={{\displaystyle \int }}_{\sigma \left(A\right)}{t}^j{d}_{E_A}=A^j, j\in \mathbb{N}.$$

To prove $\left(b\right)\Rightarrow \left(a\right),$ we use the corresponding implication of Theorem 6, where $X$ stands for $C\left(\sigma \left(A\right)\right),x_j$ stands for ${\phi }_j, j\in \mathbb{N}, Y$ stands for $Y\left(A\right), y_j$ stands for $B_j, j\in \mathbb{N},$ and $P\left(f\right)≔{\int }_{\sigma \left(A\right)}\left|f\left(t\right)\right|d_{E_A} \forall f\in C\left(\sigma \left(A\right)\right)$ defines a sublinear continuous operator, which is symmetric, that is $P\left(f\right)=P\left(-f\right)$ $\forall f\in C\left(\sigma \left(A\right)\right)$. Since ${\lambda }_j$ from (b) of Theorem 6 are non-negative, and $t^j\ge 0, t\in \sigma \left(A\right), j\in \mathbb{N},$ from the assumption (b), for any finite subset $J_0\subset \mathbb{N},$ we infer that

$$B_j\le A^j\Rightarrow {\lambda }_j{B}_j\le {\lambda }_j{A}^j, j\in J_0\Rightarrow {\displaystyle \sum }_{ j\in J_0}{\lambda }_j{B}_j\le {\displaystyle \sum }_{ j\in J_0}{\lambda }_j{A}^j.$$

On the other hand, ${\sum }_{j\in J_0}{\lambda }_j{\phi }_j\le f$ in $C\left(\sigma \left(A\right)\right)$ means ${\sum }_{ j\in J_0}{\lambda }_j{t}^j\le f\left(t\right)$ for all $t\in \sigma \left(A\right)$. This yield:

$${\displaystyle \sum }_{ j\in J_0}{\lambda }_j{A}^j={{\displaystyle \int }}_{\sigma \left(A\right)}\left({\displaystyle \sum }_{ j\in J_0}{\lambda }_j{t}^j\right)d_{E_A}\le {{\displaystyle \int }}_{\sigma \left(A\right)}f\left(t\right)d_{E_A}\le {{\displaystyle \int }}_{\sigma \left(A\right)}\left|f\left(t\right)\right|d_{E_A}=P\left(f\right).$$

According to Theorem 6, there exists a positive linear operator $T$ from int $Y=Y\left(A\right),$ with

$$T\left({\phi }_j\right)\ge B_j, j\in \mathbb{N}, T\left(f\right)\le {{\displaystyle \int }}_{\sigma \left(A\right)}\left|f\left(t\right)\right|d_{E_A}=P\left(f\right) \forall f\in C\left(\sigma \left(A\right)\right).$$

Consider the condition $\Vert f\Vert \le 1$, which is equivalent to $-1\le f\left(t\right)\le 1 \forall t\in \sigma \left(A\right).$ It results:

$$\pm T\left(f\right)=T\left(\pm f\right)\le {{\displaystyle \int }}_{\sigma \left(A\right)}\left|f\left(t\right)\right|d_{E_A}\le {{\displaystyle \int }}_{\sigma \left(A\right)}1d_{E_A}=I.$$

This implies $\left|T\left(f\right)\right|\le I,$ and, since $Y$ is a Banach lattice, the conclusion is

$$\Vert T\left(f\right)\Vert \le \Vert I\Vert =1, f\in C\left(\sigma \left(A\right)\right), \Vert f\Vert \le 1.$$

Thus, $\Vert T\Vert \le 1, \Vert P\Vert =1$. This concludes the proof. □

3.2. Markov Moment Problems on Unbounded Subsets

As it is well known, unlike the case $n=1,$ the form of nonnegative polynomials on ${\mathbb{R}}^n$ and on ${\mathbb{R}}_{+}^n, n\ge 2,$ in terms of sums of squares is not known. This is the motivation for the next polynomial approximation results, which solve this problem for the Markov moment problem in several dimensions. The general idea is to approximate an arbitrary continuous compactly supported nonnegative function on unbounded closed subset $F$ of ${\mathbb{R}}^n$ by nonnegative polynomials on $F.$ If $F={\mathbb{R}}^n$ or $F={\mathbb{R}}_{+}^n$, the approximating polynomials are sums of polynomials defined by (1), hence they are particular cases of sums of squares of polynomials in $n$ real variables. The approximation holds in $L_{\nu }^1$ spaces, where $\nu$ is a moment determinate measure. From now on, by a determinate measure we mean a positive regular Borel moment determinate measure, with finite moments of all orders.

Lemma 3

(See [34]). Let $F\subseteq {\mathbb{R}}^n$ be an unbounded closed subset, and $\nu$ a determinate measure on $F$ (with finite moments of all natural orders). Then, for any $x\in C_c\left(F\right), x\left(t\right)\ge 0, \forall t\in F,$ there exists a sequence ${\left(p_m\right)}_m, p_m\ge x, m\in \mathbb{N}, p_m\to x$ in $L_{\nu }^1\left(F\right)$ . We have:

$$\underset{m}{\lim }\underset{F}{\overset{}{{\displaystyle \int }}}{p}_m\left(t\right)d\nu =\underset{F}{\overset{}{{\displaystyle \int }}}x\left(t\right)d\nu ,$$

${\mathcal{P}}_{+}$ is dense in ${\left(L_{\nu }^1\left(F\right)\right)}_{+}$, and $\mathcal{P}$ is dense in $L_{\nu }^1\left(F\right).$

Proof. 

To prove the assertions of the statement, it is sufficient to show that for any $x\in {\left(C_0\left(F\right)\right)}_{+}$, we have

$$Q_1\left(x\right)≔inf\left\{{{\displaystyle \int }}_{F}p\left(t\right)d\nu ;p\ge x, p\in \mathcal{P}\right\}={{\displaystyle \int }}_{F}x\left(t\right)d\nu .$$

Obviously, one has

$${\mathrm{Q}}_1\left(\mathrm{x}\right)\ge {{\displaystyle \int }}_{\mathrm{F}}\mathrm{x}\left(\mathrm{t}\right)\mathrm{d}\mathsf{\nu }$$

To prove the converse, we define the linear form

$$T_0:X_0≔\mathcal{P}\oplus Sp\left\{x\right\}\to \mathbb{R}, F_0\left(p+\alpha x\right)≔{{\displaystyle \int }}_{F}p\left(t\right)d\nu +\alpha Q_1\left(x\right), p\in \mathcal{P}, \alpha \in \mathbb{R}.$$

Next, we show that $F_0$ is positive on $X_0$. In fact, for $\alpha <0$, one has (from the definition of $Q_1$, which is a sublinear functional on

$$X_1≔\left\{f\in L_{\nu }^1\left(F\right); \exists p\in \mathcal{P}, \left|f\left(t\right)\right|\le p\left(t\right) \forall t\in F\right\},$$

The following implications hold.

$$p+\alpha x\ge 0\Rightarrow p\ge -\alpha x\Rightarrow \left(-\alpha \right)Q_1\left(x\right)=Q_1\left(-\alpha x\right)\le \underset{F}{\overset{}{{\displaystyle \int }}}p\left(t\right)d\nu \Rightarrow T_0\left(p+\alpha x\right)\ge 0.$$

If $a\ge 0$, we infer that:

$$0=Q_1\left(0\right)=Q_1\left(\alpha x-\alpha x\right)\le \alpha Q_1\left(x\right)+Q_1\left(-\alpha x\right)\Rightarrow$$

$$\underset{F}{\overset{}{{\displaystyle \int }}}p\left(t\right)d\nu \ge Q_1\left(-\alpha x\right)\ge -\alpha Q_1\left(x\right)\Rightarrow T_0\left(p+\alpha x\right)\ge 0$$

where, in both possible cases, we have $x_0\in {\left(X_0\right)}_{+}\Rightarrow T_0\left(x_0\right)\ge 0$. Since $X_0$ contains the space of the polynomials’ functions, which is a majorizing subspace of $X_1$, there exists a linear positive extension $T:X_1\to \mathbb{R}$ of $T_0,$ which is continuous on $C_c\left(F\right)$ with respect to the sup-norm. Therefore, $T$ has a representation by means of a positive Borel regular measure $\mu$ on $F$, such that

$$T\left(x\right)={{\displaystyle \int }}_{F}x\left(t\right)d\mu , x\in C_c\left(F\right).$$

Let $p\in {\mathcal{P}}_{+}$ be a non-negative polynomial function. There is a nondecreasing sequence ${\left(x_m\right)}_m$ of continuous non-negative function with compact support, such that $x_m\nearrow p$ pointwise on $F$. Positivity of $T$ and Lebesgue’s dominated convergence theorem for $\mu$ yield

$$\underset{F}{\overset{}{{\displaystyle \int }}}p\left(t\right)d\nu =T\left(p\right)\ge supT\left(x_m\right)=sup{{\displaystyle \int }}_F{x}_m\left(t\right)d\mu ={{\displaystyle \int }}_{F}p\left(t\right)d\mu , p\in {\mathcal{P}}_{+}.$$

Thanks to Haviland’s theorem, there exists a positive Borel regular measure $\lambda$ on $F$, such that

$$\lambda \left(p\right)=\nu \left(p\right)-\mu \left(p\right)\iff \nu \left(p\right)=\lambda \left(p\right)+\mu \left(p\right), p\in \mathcal{P}.$$

Since $\nu$ is assumed to be M-determinate, it follows that

$$\nu \left(B\right)=\lambda \left(B\right)+\mu \left(B\right)$$

for any Borel subset $B$ of $F$. From this last assertion, approximating each $x\in {\left(L_{\nu }^1(F\right))}_{+}$, by a nondecreasing sequence of non-negative simple functions, and using Lebesgue’s convergence theorem, one obtains firstly for positive functions, then for arbitrary $\nu$-integrable functions, $\phi :$

$${{\displaystyle \int }}_F\phi d\nu ={{\displaystyle \int }}_F\phi d\lambda +{{\displaystyle \int }}_F\phi d\mu , \phi \in L_{\nu }^1\left(F\right).$$

In particular, we must have

$${{\displaystyle \int }}_{F}xd\nu \ge {{\displaystyle \int }}_{F}xd\mu =T\left(x\right)=T_0\left(x\right) =Q_1\left(x\right).$$

The conclusion follows. □

Remark 1.

We recall that the preceding Lemma 3 is no more valid when we replace$L_{\nu }^1\left(F\right)$with the Hilbert space$L_{\nu }^2\left(F\right), F={\mathbb{R}}^n, n\ge 2$(see [23], where the authors construct such a measure$\nu$).

Lemma 4

(See [37]). Let $\nu ={\nu }_1\times \cdots \times {\nu }_n$ be a product of n determinate measures on $\mathbb{R}$ . Then we can approximate any nonnegative continuous compactly supported function $\psi \in X={\left(C_c\left({\mathbb{R}}^n\right)\right)}_{+}$ with sums of products

$$\begin{gathered} p_1\otimes \cdots \otimes p_n, \\ (p_1\otimes \cdots \otimes p_n)\left(t_1,\dots ,t_n\right)≔p_1\left(t_1\right)\cdots p_n\left(t_n\right)·t=\left(t_1,\dots ,t_n\right)\in {\mathbb{R}}^n, \end{gathered}$$

$p_j$ nonnegative polynomial on the entire real line, $j=1,\dots ,n$, and any such sum of special polynomials dominates $\psi$ on ${\mathbb{R}}^n.$

Lemma 5

(See [37]). Let $\nu ={\nu }_1\times \cdots \times {\nu }_n$ be a product of n determinate measures on ${\mathbb{R}}_{+},$ Then, we can approximate any nonnegative continuous compactly supported function $\psi \in {\left(C_c\left({\mathbb{R}}_{+}^n\right)\right)}_{+}$ with sums of products

$$\begin{gathered} p_1\otimes \cdots \otimes p_n, \\ (p_1\otimes \cdots \otimes p_n)\left(t_1,\dots ,t_n\right)≔p_1\left(t_1\right)\cdots p_n\left(t_n\right)·t=\left(t_1,\dots ,t_n\right)\in {\mathbb{R}}_{+}^n, \end{gathered}$$

$p_j$ nonnegative polynomial on the entire nonnegative semi axes, $j=1,\dots ,n$, and any such sum of special polynomials dominates $\psi$ on ${\mathbb{R}}_{+}^n.$

Example 2.

(i) The measure

$$exp\left(-{{\displaystyle \sum }}_{j=1}^n{\alpha }_j{t}_j^2\right)d{t}_1\cdots d{t}_n=\left(e^{-{\alpha }_1{t}_1^2}d{t}_1\right)\times \cdots \times \left(e^{-{\alpha }_n{t}_n^2}d{t}_n\right), {\alpha }_j>0, j=1,\dots ,n,$$

is moment determinate on${\mathbb{R}}^n.$Indeed, for$n=1, \alpha >0,$the measure$e^{-t^2}dt$is moment determinate on$\mathbb{R},$according to [24]. For$n\ge 2$the product of determinate measures on$\mathbb{R}$is a determinate measure$\nu$on${\mathbb{R}}^n,$since the polynomials are dense in$L_{\nu }^1\left({\mathbb{R}}^n\right),$according to Lemma 4 and measure theory results.

(ii) For any${\alpha }_j>0, j=1,\dots ,n,$the measure

$$exp\left(-{{\displaystyle \sum }}_{j=1}^n{\alpha }_j{t}_j\right)d{t}_1\cdots d{t}_n$$

is moment determinate on${\mathbb{R}}_{+}^n$, because of similar reasons to those mentioned at point (i), accompanied by Lemma 5.

Application of Lemma 3 and other appropriate results lead to the following theorem, which holds on arbitrary unbounded (or bounded) closed subsets.

Theorem 8.

Let$F$be a closed unbounded subset of${\mathbb{R}}^n,$$Y$an order complete Banach lattice,${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$a given sequence in$Y,$$\nu$a determinate measure on$F.$Let$T_{1,}{T}_2\in B\left(L_{\nu }^1\left(F\right),Y\right)$two linear bounded operators from$L_{\nu }^1\left(F\right)$into$Y.$The following statements are equivalent:

(a)

there exists a unique bounded linear operator$T\in B\left( L_{\nu }^1\left(A\right),Y\right)$such that$T\left({\phi }_j\right)=y_j, j\in {\mathbb{N}}^n,$$T$is between$T_1$and$T_2$on the positive cone of$L_{\nu }^1\left(F\right).$

(b)

For any finite subset$J_0\subset {\mathbb{N}}^n,$and any${\left\{a_j\right\}}_{j\in J_0}\subset \mathbb{R},$we have

$${\sum }_{j\in J_0}{a}_j{\phi }_j\ge 0 \mathit{on} F\Rightarrow$$

$${\displaystyle \sum }_{j\in J_0}{a}_j{T}_1\left({\phi }_j\right)\le {\displaystyle \sum }_{j\in J_0}{a}_j{y}_j\le {\displaystyle \sum }_{j\in J_0}{a}_j{T}_2\left({\phi }_j\right)$$

In the cases when the analytic form of nonnegative polynomials in terms of sums of squares is known, Theorem 8 solves the existence and uniqueness of the solution for the full Markov moment problem in terms of quadratic forms. Here are a few such examples, formulated as consequences of Theorem 8. The simplest cases are $F=\mathbb{R}$ and $F={\mathbb{R}}_{+}$, regarded as closed subsets of $\mathbb{R}.$

Corollary 5.

Let$X=L_{\nu }^1\left(\mathbb{R}\right),$where$\nu$is a determinate measure on$\mathbb{R}.$Assume that$Y$is an arbitrary order complete Banach lattice, and${\left(y_n\right)}_{n\ge 0}$is a given sequence with its terms in$Y.$Let$T_1,T_2$be two linear operators from$X$to$Y$, such that$\mathbf{0}\le T_1\le T_2$on$X_{+}.$The following statements are equivalent:

(a)

There exists a unique bounded linear operator$T$from$X$to$Y,$$T_1\le T\le T_2$on$X_{+}, \Vert T_1\Vert \le \Vert T\Vert \le \Vert T_2\Vert$, such that$T\left({\phi }_n\right)=y_n$for all$n\in \mathbb{N};$

(b)

If$J_0\subset \mathbb{N}$is a finite subset, and$\left\{{\lambda }_j;j\in J_0\right\}\subset \mathbb{R},$then

$${\displaystyle \sum }_{i,j\in J_0}^{}{\lambda }_i{\lambda }_j{T}_1\left({\phi }_{i+j}\right)\le {\displaystyle \sum }_{i,j\in J_0}^{}{\lambda }_i{\lambda }_j{y}_{i+j}\le {\displaystyle \sum }_{i,j\in J_0}^{}{\lambda }_i{\lambda }_j{T}_2\left({\phi }_{i+j}\right).$$

Corollary 6.

Let$X=L_{\nu }^1\left({\mathbb{R}}_{+}\right),$where$\nu$is a moment-determinate measure on${\mathbb{R}}_{+}.$Assume that$Y$is an arbitrary order complete Banach lattice, and${\left(y_n\right)}_{n\ge 0}$is a given sequence with its terms in$Y.$Let$T_1,T_2$be two linear operators from$X$to$Y$, such that$\mathbf{0}\le T_1\le T_2$on$X_{+}$. As usual, we denote${\phi }_j\left(t\right)=t^j, j\in \mathbb{N}, t\in {\mathbb{R}}_{+}$. The following statements are equivalent:

(a)

There exists a unique bounded linear operator$T$from$X$to$Y, T_1\le T\le T_2$on$X_{+},$$\Vert T_1\Vert \le \Vert T\Vert \le \Vert T_2\Vert$, such that$T\left({\phi }_n\right)=y_n$for all$n\in \mathbb{N};$

(b)

If$J_0\subset \mathbb{N}$is a finite subset, and$\left\{{\lambda }_j;j\in J_0\right\}\subset \mathbb{R},$then

$${\displaystyle \sum }_{i,j\in J_0}^{}{\lambda }_i{\lambda }_j{T}_1\left({\phi }_{i+j+k}\right)\le {\displaystyle \sum }_{i,j\in J_0}^{}{\lambda }_i{\lambda }_j{y}_{i+j+k}\le {\displaystyle \sum }_{i,j\in J_0}^{}{\lambda }_i{\lambda }_j{T}_2\left({\phi }_{i+j+k}\right), k\in \left\{0,1\right\}.$$

Example 3.

Let$d\nu ≔e^{-t}dt, t\in {\mathbb{R}}_{+}, X=L_{\nu }^1\left({\mathbb{R}}_{+}\right),$and${\left(y_n\right)}_{n\ge 0}$a sequence of real numbers. The following statements are equivalent:

(a)

There exists$f\in L_{\nu }^{\infty }\left({\mathbb{R}}_{+}\right), t^2{e}^{-t}\le f\left(t\right)\le 1$for almost all$t\in {\mathbb{R}}_{+}$, such that

$${{\displaystyle \int }}_0^{\infty }{t}^{j}f\left(t\right)e^{-t}dt=y_j, \forall j\in \mathbb{N}.$$

(b)

If$J_0\subset \mathbb{N}$is a finite subset, and$\left\{{\lambda }_j;j\in J_0\right\}\subset \mathbb{R}$, then

$${\displaystyle \sum }_{i,j\in J_0}^{}{\lambda }_i{\lambda }_j\frac{\left(2+i+j+k\right)!}{2^{3+i+j+k}}\le {\displaystyle \sum }_{i,j\in J_0}^{}{\lambda }_i{\lambda }_j{y}_{i+j+k}\le {\displaystyle \sum }_{i,j\in J_0}^{}{\lambda }_i{\lambda }_j\left(i+j+k\right)!, k\in \left\{0,1\right\}$$

To obtain the equivalence from Example 3, one applies Corollary 6 to $Y=\mathbb{R},$ also using measure theory arguments [9] and properties of the Gamma function.

For $n\ge 2,$ the expression of nonnegative polynomials on ${\mathbb{R}}^n$ or on ${\mathbb{R}}_{+}^n$ in terms of sums of squares is not known. A way of avoiding this difficulty is to apply approximation Lemmas 4 and 5, supplied by the following general theorem.

Theorem 9

(See [34]). Let $F\subseteq {\mathbb{R}}^n$ be a closed unbounded subset; $\nu$ a moment-determinate measure on $F,$ having finite moments of all orders; and $X=L_{\nu }^1\left(F\right), {\phi }_j\left(t\right)=t^j, t\in F, j\in {\mathbb{N}}^n.$ Let $Y$ be an order complete Banach lattice, ${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$ a given sequence of elements in $Y, T_1$ and $T_2$ two bounded linear operators from $X$ to $Y.$ Assume that there exists a sub-cone ${\mathcal{P}}_{++}\subseteq {\mathcal{P}}_{+},$ such that each $f\in {\left(C_c\left(F\right)\right)}_{+}$ can be approximated in$X$ by a sequence ${\left(p_l\right)}_l, p_l\in {\mathcal{P}}_{++}, p_l\ge f$ for all $l$. The following statements are equivalent:

(a)

There exists a unique (bounded) linear operator$T:X\to Y, T\left({\phi }_j\right)=y_j, j\in {\mathbb{N}}^n, \mathbf{0}\le T_1\le T\le T_2$on$X_{+}, \Vert T_1\Vert \le \Vert T\Vert \le \Vert T_2\Vert .$

(b)

For any finite subset$J_0\subset {\mathbb{N}}^n,$and any$\left\{{\lambda }_j;j\in J_0\right\}\subset \mathbb{R},$the following implications hold true:

$${\displaystyle \sum }_{j\in J_0}{\lambda }_j{\phi }_j\in {\mathcal{P}}_{+}\Rightarrow {\displaystyle \sum }_{j\in J_0}{\lambda }_j{T}_1\left({\phi }_j\right)\le {\displaystyle \sum }_{j\in J_0}{\lambda }_j{y}_j,$$

$${\displaystyle \sum }_{j\in J_0}{\lambda }_j{\phi }_j\in {\mathcal{P}}_{++}\Rightarrow {\displaystyle \sum }_{j\in J_0}{\lambda }_j{T}_1\left({\phi }_j\right)\ge 0, {\displaystyle \sum }_{j\in J_0}{\lambda }_j{y}_j \le {\displaystyle \sum }_{j\in J_0}{\lambda }_j{T}_2\left({\phi }_j\right).$$

When $F={\mathbb{R}}^n,$ the convex cone ${\mathcal{P}}_{++}\left({\mathbb{R}}^n\right)$ is the set of all finite sums of polynomials defined by (1), where each $p_j$ is nonnegative on the entire real axis. Thus $p_j=q_j^2+r_j^2$ for some $q_j,r_j\in \mathbb{R}\left[t\right], j=1,\dots ,n.$ From Theorem 9 and Lemma 4 we obtain:

Theorem 10.

Let$\nu ={\nu }_1\times \cdots \times {\nu }_n, n\ge 2,{\nu }_j$being a determinate measure on$\mathbb{R}, j=1,\dots ,n, X=L_{\nu }^1\left({\mathbb{R}}^n\right), {\phi }_j\left(t\right)=t^j, t\in {\mathbb{R}}^n, j\in {\mathbb{N}}^n.$Let$Y$be an order complete Banach lattice,${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$a given sequence of elements in$Y, and T_1$and$T_2$two bounded linear operators mapping$X$into$Y.$The following statements are equivalent:

(a)

There exists a unique (bounded) linear operator

$$T:X\to Y, T\left({\phi }_j\right)=y_j, j\in {\mathbb{N}}^n, \mathbf{0}\le T_{\mathbf{1}}\le T\le T_2 on X_{+}, \Vert T_1\Vert \le \Vert T\Vert \le \Vert T_2\Vert .$$

(b)

For any finite subset$J_0\subset {\mathbb{N}}^n,$and any$\left\{{\lambda }_j;j\in J_0\right\}\subset \mathbb{R}$, the following implication holds true:

$${\displaystyle \sum }_{j\in J_0}{\lambda }_j{\phi }_j\in {\mathcal{P}}_{+}\Rightarrow {\displaystyle \sum }_{j\in J_0}{\lambda }_j{T}_1\left({\phi }_j\right)\le {\displaystyle \sum }_{j\in J_0}{\lambda }_j{y}_j.$$

For any finite subsets$J_k\subset \mathbb{N}, k=1,\dots ,n,$and any${\left\{{\lambda }_{j_k}\right\}}_{j_k\in J_k}\subset \mathbb{R}$, the following inequalities hold true:

$$\mathbf{0}\le {\displaystyle \sum }_{i_1,j_1\in J_1}\left(\cdots \left({\displaystyle \sum }_{i_n.j_n\in J_n}{\lambda }_{i_1}{\lambda }_{j_1}\cdots {\lambda }_{i_n}{\lambda }_{j_n}{T}_1\left({\phi }_{i_1+j_1,\dots ,i_n+j_n}\right)\right)\cdots \right),$$

$${\displaystyle \sum }_{i_1,j_1\in J_1}\left(\cdots \left({\displaystyle \sum }_{i_n.j_n\in J_n}{\lambda }_{i_1}{\lambda }_{j_1}\cdots {\lambda }_{i_n}{\lambda }_{j_n}{y}_{i_1+j_1,\dots ,i_n+j_n}\right)\cdots \right)\le$$

$${\displaystyle \sum }_{i_1,j_1\in J_1}\left(\cdots \left({\displaystyle \sum }_{i_n.j_n\in J_n}{\lambda }_{i_1}{\lambda }_{j_1}\cdots {\lambda }_{i_n}{\lambda }_{j_n}{T}_2\left({\phi }_{i_1+j_1,\dots ,i_n+j_n}\right)\right)\cdots \right).$$

If $F={\mathbb{R}}_{+}^n, n\ge 2,$ then ${\mathcal{P}}_{++}\left({\mathbb{R}}_{+}^n\right)$ will be the convex cone generated by the polynomials defined by (1), where each $p_j$ is nonnegative on ${\mathbb{R}}_{+},$ hence $p_j\left(t\right)=q_j^2\left(t\right)+t{r}_j^2\left(t\right), t\in {\mathbb{R}}_{+},$ for some $q_j,r_j\in \mathbb{R}\left[t\right], j=1,\dots ,n$.

4. Discussion

This paper provides solutions for the one dimensional and for the multidimensional Markov moment problems over the real field, essentially based on polynomial approximation on the closed subset $F$ of ${\mathbb{R}}^n.$ It is based on results in functional analysis over the real field. When $F=K$ is a compact with non-empty interior or a semi-algebraic compact, uniform approximation holds since the approximation is done in the space $C\left(K\right)$. In the case of an unbounded closed subset $F,$ polynomial approximation in $L_{\nu }^1\left(F\right)$ is proved and applied, where $\nu$ is a moment determinate measure on $F.$ Thus, we can say that polynomial approximation solves full Markov moment problems on bounded, as well as on unbounded special closed subsets of ${\mathbb{R}}^n$. In the case of an unbounded closed subset $F,$ polynomial approximation in $L_{\nu }^1\left(F\right)$ is proved and applied, where $\nu$ is a determinate positive regular Borel measure on $F,$ with finite moments of all orders. In this case, for any nonnegative function $\psi \in C_c\left(F\right)$, the approximating polynomials dominate $\psi .$ E few examples are given, For $F$ = ${\mathbb{R}}^n$ and $F={\mathbb{R}}_{+}^n$, and $\nu ={\nu }_1\times \cdots \times {\nu }_n,$ each ${\nu }_j$ being determinate on $\mathbb{R},$ or on ${\mathbb{R}}_{+}, j=1,\dots ,n,$ the involved polynomials can be expressed in terms of sums of squares. Our results hold for the vector valued Markov moment problem, the codomain being an arbitrary order complete Banach lattice. In the one-dimensional case, for compact subsets, the elements of this Banach lattice are self-adjoint (symmetric) operators acting on a real or complex Hilbert space. The interested reader can complete the information in this work with the aid of the references, making the connections with other topics, such as operator theory, algebra (especially matrix theory), geometry, optimization, Fourier series and their partial sums, uniqueness, and construction of the solution for moment problems. In some cases, a reduced (truncated) moment problem and related problems mentioned above can be solved by means of a finite number of operations.

5. Conclusions

The main results on polynomial approximation on unbounded closed subsets are reviewed in Lemmas 3, 4 and 5. They lead to characterizations of the existence and uniqueness of the solution for Markov moment problems stated in Theorems 8, 9 10 and their corollaries. The case of Markov moment problem on large classes of compact subsets is covered by the results of Section 3.1. The notion of symmetric operator acting on a Hilbert space appear naturally in the definition of the order complete Banach lattice defined by (10), in Corollary 1, Theorem 7. The same Theorem 7 uses the sublinear operator $P$, which satisfies another type of symmetry condition, namely $P\left(f\right)=P\left(-f\right)$ for all $f$ in the domains space. A scalar valued Markov moment problem is solved in Corollary 6. Thus, the three aims outlined in the Abstract are covered. As a possible direction for future work, we mention application of the above results to concrete codomain spaces (such as spaces of symmetric square matrices with real entries defined by (10), function spaces and other appropriate codomain spaces for which the theory works).