Approximation and the Multidimensional Moment Problem

Abstract

The aim of this paper is to apply polynomial approximation by sums of squares in several real variables to the multidimensional moment problem. The general idea is to approximate any element of the positive cone of the involved function space with sums whose terms are squares of polynomials. First, approximations on a Cartesian product of intervals by polynomials taking nonnegative values on the entire ${\mathbb{R}}^2$, or on ${\mathbb{R}}_{+}^2$, are considered. Such results are discussed in $L_{\mu }^1\left({\mathbb{R}}^2\right)$ and in $C\left(S_1\times S_2\right)$-type spaces, for a large class of measures, $\mu ,$ for compact subsets $S_i, i=1,2$ of the interval $[0,+\infty )$. Thus, on such subsets, any nonnegative function is a limit of sums of squares. Secondly, applications to the bidimensional moment problem are derived in terms of quadratic expressions. As is well known, in multidimensional cases, such results are difficult to prove. Directions for future work are also outlined.

1. Introduction

We recall that given a multi-indexed sequence ${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$ of real numbers, by solving the $n$-dimensional full moment problem on the closed subset $S\subseteq {\mathbb{R}}^n$, usually it means finding necessary and sufficient conditions on the sequence ${\left(y_j\right)}_{j\in {\mathbb{N}}^n},$ for the existence and uniqueness of a positive Borel measure $\mu$ on $S$, whose moments ${\int }_S{t}^{j}d\mu$ have the prescribed values $y_j$ for all $j\in {\mathbb{N}}^n.$ Thus, the following equalities are required:

$${\int }_S{t}^{j}d\mu =y_j, j\in {\mathbb{N}}^n.$$

(1)

In this study, we focus on the more specific problem. Namely, let $S$ be a closed unbounded subset of ${\mathbb{R}}^n, n\ge 2$, $X$ a Banach sublattice in ${\mathbb{R}}^S$, $Y$ a Banach lattice, and $T_2$ a positive linear operator from $X$ to $Y$. Let ${\left(y_j\right)}_{j\in {\mathbb{N}}^n}$ be given a sequence of elements in $Y.$ The problem is to characterize the existence and uniqueness of a linear operator $T:X⟶Y$, verifying the moment conditions

$$T\left(p_j\right)=y_j, j\in {\mathbb{N}}^n,$$

(2)

and the sandwich inequalities

$$0\le T\left(\phi \right)\le T_2\left(\phi \right), \forall \phi \in X_{+}.$$

(3)

on the positive cone $X_{+}$ of the domain space $X$.

The following notations have been used:

$$\mathbb{N}≔\left\{0,1,2,\dots \right\},{\mathrm{p}}_{\mathrm{j}}\left(\mathrm{t}\right)≔{\mathrm{t}}^{\mathrm{j}}≔{\mathrm{t}}_1^{{\mathrm{j}}_1}·\cdots ·{\mathrm{t}}_{\mathrm{n}}^{{\mathrm{j}}_{\mathrm{n}}},\mathrm{j}=\left({\mathrm{j}}_1,\dots ,{\mathrm{j}}_{\mathrm{n}}\right)\in {\mathbb{N}}^{\mathrm{n}},\mathrm{t}=\left({\mathrm{t}}_1,\dots ,{\mathrm{t}}_{\mathrm{n}}\right)\in \mathrm{S}.$$

For general related knowledge regarding the analysis, see the references [1,2,3,4,5,6,7,8]. The one-dimensional case $n=1, S≔\mathbb{R}$ and, respectively, $S={\mathbb{R}}_{+}≔\left[0,+\infty \right)$ were the first problems that have been solved. This is possible due to the explicit form of decomposition of nonnegative polynomials on $\mathbb{R}$, respectively, on ${\mathbb{R}}_{+}$, as sums of squares, respectively, as sums of polynomials of the form

$$t^s{q}^2\left(t\right), q\in \mathbb{R}\left[t\right], t\in {\mathbb{R}}_{+}, s\in \left\{0,1\right\}.$$

(4)

This problematic has been reviewed in [8]. Unlike the one-dimensional case of nonnegative polynomials on $\mathbb{R}$, as is known [9], there exist polynomials having nonnegative values at all points in ${\mathbb{R}}^2$, that are not sums of squares. In the same article, the authors prove that there are positively definite sequences of vectors in ${\mathbb{R}}^n$, $n\ge 2,$ which are not moment sequences. Therefore, it seems that solving the multidimensional moment problem in terms of quadratic forms and sums of squares of polynomials encounters difficulties. Another solved moment problem was that over compact subsets $K$ of ${\mathbb{R}}^n, n\ge 1$, and especially the case $n\ge 2.$ In this respect, important results have been published or/and reviewed in [10,11,12,13,14]. Moreover, the decomposition of polynomials taking positive values at all points of such subsets in terms of polynomials defining the compact $K$ has been established. In [15], an operator valued moment problem has been solved. Going to the case of unbounded closed subsets $S\subseteq {\mathbb{R}}^n, n\ge 2,$ for solving $n$-dimensional moment problems in $L_{\mu }^1\left(S\right)$ spaces, the notion of a determinate measure $\mu$ on $S$ plays an essential role. Whether $\mu$ is the product of $n$-determinate measures on $\mathbb{R}$ and, respectively, on ${\mathbb{R}}_{+}$ is of particular interest. Firstly, in [16], the checkable criteria for the determinacy of measures on $\mathbb{R}$ and on ${\mathbb{R}}_{+}$ are proved. References [17,18,19] provide recent results on the moment problem, some of them involving entropy and fractional moments. In the articles [20,21], a connection between the moment problem and operator theory is pointed out. The article [22] is not directly related to the moment problem. However, in the present paper, we use the fact that any positive linear operator on Banach lattices is continuous. A more general result, on positive linear operators acting between ordered Banach spaces, is recalled and proved in [22]. The articles [23,24] refer to expansions motivated by applications. An approximation of functions can also be made with Bernstein-type operators emphasized in the literature (see the references [25,26,27,28]). References [29,30,31,32] contain results on the moment problem and related approximation directly applied in the present work. We recall that if $\mu ={\mu }_1\times \cdots \times {\mu }_n$ and each ${\mu }_i$ is determinate on $\mathbb{R},$ then the density of nonnegative polynomials in ${\left(L_{{\mu }_i}^1\left(\mathbb{R}\right)\right)}_{+}$, $i=1,\dots ,n$ leads to the density of the space generated by special nonnegative polynomials

$$q\left(t\right)=\left(q_1⨂\cdots ⨂q_n\right)\left(t\right)≔q_1\left(t_1\right)\cdots q_n\left(t_n\right), t\in {\mathbb{R}}^n,q_i \in {\left(\mathbb{R}\left[t_i\right]\right)}_{+}, i=1,\dots ,n.$$

(5)

in ${\left(L_{\mu }^1\left({\mathbb{R}}^n\right)\right)}_{+}$. Since each $q_i$ is a sum of squares in the variable $t_i$, our requirements can be characterized in terms of products of quadratic forms. Another approximation-type result used in this paper is that of uniform approximation of any continuous nonnegative function on a compact subset of $K\subset {\mathbb{R}}_{+}$ by restrictions to $K$ of nonnegative polynomials on the entire interval $\left[0,+\infty \right)$. A corresponding result for the Cartesian product of compact subsets $S=S_1\times S_2\subset {[0,+\infty )}^2$ is going to be applied in solving the moment problems proved in Section 3.2 below. The rest of the paper is organized as follows. Section 2 summarizes the methods used in the article. Section 3 contains the results of the study. Section 4 discusses some of the results, and Section 5 concludes the paper. Generally (but not always), the articles from the references has been numbered in chronological order of their publication date.

2. Methods

Here are the most used methods applied in this work.

• The approximation of nonnegative functions on closed unbounded subsets $S$ of ${\mathbb{R}}^n$ endowed with a moment determinate measure $\mu ,$ in $L_{\mu }^1\left(S\right),$ by nonnegative polynomials on $S$ (see [31], Lemma 4.11, p. 383 and the references therein). Proving the corresponding approximation result in $L_{\mu }^1\left({\mathbb{R}}^2\right),$ for a product measure $\mu ={\mu }_1\times {\mu }_2$, where ${\mu }_i$ is moment determinate on $\mathbb{R},i=1,2.$
• Approximation of nonnegative continuous functions on a compact subset $S$ of $[0,+\infty )$ by nonnegative polynomials on the entire demi axes $\left[0,+\infty \right)$ in the space $C\left(S\right)$ (Lemma 2 from [30], p. 5). Deducing the corresponding result for approximation on compact subsets $S=S_1\times S_2\subset {[0,+\infty )}^2$ (see Section 3.1 below).
• Using ideas on the moment problem and the form of positive polynomials over $\mathbb{R}$ (Hamburger moment problem), over ${\mathbb{R}}_{+}$ (Stieltjes moment problem), and over special compact subsets of ${\mathbb{R}}^n, n\ge 1$ [8,9,10,11,12,13,14].
• Using the fact that any positive linear operator acting on ordered Banach spaces is continuous (see Lemma 2 from [22], p. 2).
• Applying measure theory results [1,2] (for example, see [2], pp. 5–75, 116–177).
• Using the result on self-adjoint operators acting on Hilbert space $H$ and a commutative Banach algebra of such operators. This algebra $Y$ is also a Banach lattice with respect to the usual order relation on the real vector $Y_0$ of all (bounded) self-adjoint operators, which is defined by $U\le V$ if and only if $⟨Uh,h⟩\le ⟨Vh,h⟩$ for all $h\in H$ and any $U,V\in Y_0$. With respect to this order relation, the ordered vector space $Y_0$ is not a lattice. However, the commutative algebra $Y$ mentioned above is an order complete Banach lattice [5]. However, in the present paper, it seems we are not using the order completeness of this Banach lattice. An interesting property of $Y$ is the following one: $\left|U\right|=sup\left\{U,-U\right\}=\sqrt{U^2}$ for all $U\in Y.$ If $H={\mathbb{R}}^n, n\ge 2,$ the vector ordered Banach space $Y_0$ is isomorphic to the space of all symmetric $n\times n$ matrices with real entries. The construction of the Banach lattice $Y$ is a particular case of that made for arbitrary Hilbert spaces. Here, the operatorial norm of a positive definite symmetric matrix $A$ equals its greatest eigenvalue.
• Using ideas from [3] (Theorem 12.24, pp. 325–326), and [4,5,8], regarding functional calculus for commuting self-adjoint operators.

3. Results

3.1. Polynomial Approximation

Theorem 1.

Let ${\mu }_1,{\mu }_2$ be positive regular Borel moment determinate measures on  $\mathbb{R},$, with finite absolute moments  ${\int }_{\mathbb{R}}{\left|t\right|}^{n}d{\mu }_i\left(t\right)$ of all natural orders,  $n\in \mathbb{N},i=1,2$. In addition, assume that $\mu ≔{\mu }_1\times {\mu }_2$ is the product measure on ${\mathbb{R}}^2$ of the measures ${\mu }_1$ and ${\mu }_2.$ Also, assume that $d{\mu }_i={\rho }_i\left(t_i\right)d{t}_i,$ and ${\rho }_i$ is continuous and positive on $\mathbb{R},i=1,2$. Then, the following approximations hold true.

(i)

Any function $\phi$ from ${\left(L_{{\mu }_1\times {\mu }_2}^1\left({\mathbb{R}}^2\right)\right)}_{+}$ can be approximated in the norm of $L_{{\mu }_1\times {\mu }_2}^1\left({\mathbb{R}}^2\right)$ by polynomials from the sub-cone $C\subset {\left(L_{{\mu }_1\times {\mu }_2}^1\left({\mathbb{R}}^2\right)\right)}_{+}$ generated by special polynomials $r_1\otimes r_2$, $r_i\in \mathbb{R}\left[t_i\right], r_i\left(t_i\right)\ge 0$ for all $t_i\in \mathbb{R},i=1,2$.

Consequently, we have the following:

(ii)

Any function $\phi \in {\left(L_{{\mu }_1\times {\mu }_2}^1\left({\mathbb{R}}^2\right)\right)}_{+}$ can be approximated by sums of squares whose terms have the following form:

$$q\left(t_1,t_2\right)=q_1^2\left(t_1\right)·q_2^2\left(t_2\right)=\left(q_1^2\otimes q_2^2\right)\left(t_1,t_2\right),\hspace{1em}\left(t_1,t_2\right)\in {\mathbb{R}}^2,\hspace{1em}{q}_i\in \mathbb{R}\left[t_i\right], i=1,2.$$

(6)

Proof. 

The general idea of the proof is to point out that the convex cone generated by all special polynomials of the form $\left(q_1^2\otimes q_2^2\right)\left(t_1,t_2\right)≔q_1^2\left(t_1\right)q_2^2\left(t_2\right),$ with $q_i\in \mathbb{R}\left[t_i\right], i=1,2$ is dense in the positive cone of ${\left(L_{{\mu }_1\times {\mu }_2}^1\left({\mathbb{R}}^2\right)\right)}_{+}$ of the Banach lattice $L_{{\mu }_1\times {\mu }_2}^1\left({\mathbb{R}}^2\right)$. To prove this, we observe that any nonnegative continuous compactly supported function $g\in {\left(C_c\left({\mathbb{R}}^2\right)\right)}_{+}$ can be approximated by finite sums

$${\sigma }_m≔{\sigma }_{m,1}\otimes {\sigma }_{m,2}≔{\sum }_{j_1\in I_m}{g}_{1,j}\otimes ({\sum }_{j_2\in J_m}{g}_{2,j_m})⟶g,\hspace{1em}m⟶\infty ,$$

(7)

with $g_{1,j_m}$ and $g_{2,j_m}$ continuous, nonnegative and compactly supported functions from ${\left(C_c\left(\mathbb{R}\right)\right)}_{+},$ such that $Supp\left(g_{i,j}\right)$ is contained in a interval containing ${pr}_i\left(Supp\left(g\right)\right), i=1,2, j\in J_m, m\in \mathbb{N}$ (see [1,2], and apply the Stone–Weierstrass theorem on the compact subset ${pr}_1\left(Supp\left(g\right)\right)\times {pr}_2\left(Supp\left(g\right)\right)\subset {\mathbb{R}}^2$. Here, $Supp\left(g\right)$ is the support of $g,$ and the obvious inclusion relation $Supp\left(g\right)\subseteq {pr}_1\left(Supp\left(g\right)\right)\times {pr}_2\left(Supp\left(g\right)\right)$ holds true). Here, $I_m$ and $J_m$ are both finite subsets of $\mathbb{N}$. For $i\in \left\{1,2\right\},$ since ${\mu }_i$ is moment determinate and ${\sigma }_{m,i}$ takes nonnegative values on $\mathbb{R},$ according to Lemma 4.11 from [31], (p. 383), for ${\sigma }_{m,i}\in {\left(C_c\left(\mathbb{R}\right)\right)}_{+},$ there exists sequences ${\left(r_{n,m_i}\right)}_{n\ge 0}$ of nonnegative polynomials on $\mathbb{R}$, such that

$$\underset{n\to \infty }{\lim }{r}_{n,m_i}={\sigma }_{m,i}, r_{n,m_i}\left(t\right)\ge {\sigma }_{m,i}\left(t\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} n\in \mathbb{N} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{a}\mathrm{l}\mathrm{l} t\in \mathbb{R}.$$

We recall that the lemma cited above says that for a moment determinate measure ${\mu }_i$ on $\mathbb{R},$ any continuous compactly supported function taking nonnegative values at any point of the real axes can be approximated by dominating polynomials in the space $L_{{\mu }_i}^1\left(\mathbb{R}\right), i=1,2$. The convergences hold in $L_{{\mu }_i}^1\left(\mathbb{R}\right), i=1,2$. Due to the form of nonnegative polynomials at any point from $\mathbb{R},$ they are sums of squares of polynomials on $\mathbb{R}$. With the previously used notations and results, any function $g\in {\left(C_c\left({\mathbb{R}}^2\right)\right)}_{+}$ can be approximated by finite sums

$${\sigma }_m={\sigma }_{m,1}\otimes {\sigma }_{m,2}=\left(\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{r}_{n,m_1}\right)\otimes \left(\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{r}_{n,m_2}\right).$$

The next problem is whether we can write

$${\sigma }_m=\underset{n\to \infty }{\lim }\left(r_{n,m_1}\otimes r_{n,m_2}\right) \mathrm{i}\mathrm{n}{L}_{{\mu }_1\times {\mu }_2}^1\left({\mathbb{R}}^2\right).$$

(8)

The following method could be applied. Namely, for an arbitrary small $\epsilon >0$, according to Fubini theorem, we have

$$\begin{array}{cc}\hfill {\iint }_{{\mathbb{R}}^2}(\left({\sigma }_{m,1}\otimes {\sigma }_{m,2}\right)& -\left(r_{n,m_1}\otimes r_{{n,m}_2}\right))\left(t_1,t_2\right)d_{{\mu }_1\times {\mu }_2}\hfill \\ & ={\iint }_{{\mathbb{R}}^2}\left({\sigma }_{m,1}\left(t_1\right)·{\sigma }_{m,2}\left(t_2\right)-r_{n,m_1}\left(t_1\right)·{\sigma }_{m,2}\left(t_2\right)\right)d_{{\mu }_1\times {\mu }_2}\hfill \\ & +{\iint }_{{\mathbb{R}}^2}\left(r_{n,m_1}\left(t_1\right)·{\sigma }_{m,2}\left(t_2\right)-r_{n,m_1}\left(t_1\right)·r_{{n,m}_2}\left(t_2\right)\right)d_{{\mu }_1\times {\mu }_2}\hfill \\ & ={\int }_{\mathbb{R}}\left({\sigma }_{m,1}\left(t_1\right)-r_{n,m_1}\left(t_1\right)\right)d_{{\mu }_1}·{\int }_{\mathbb{R}}{\sigma }_{m,2}\left(t_2\right)d{\mu }_2\hfill \\ & +{\int }_{\mathbb{R}}{r}_{n,m_1}\left(t_1\right)d{\mu }_1·{\int }_{\mathbb{R}}\left({\sigma }_{m,2}\left(t_2\right)-r_{n,m_2}\left(t_2\right)\right)d_{{\mu }_2}.\hfill \end{array}$$

Due to Equation (3), the following relations hold true:

$$\begin{array}{cc}\hfill \Vert \left({\sigma }_{m,1}\otimes {\sigma }_{m,2}\right)& -\left(r_{n,m_1}\otimes r_{{n,m}_2}\right){\Vert }_{L_{{\mu }_1\times {\mu }_2}^1\left({\mathbb{R}}^2\right)}\hfill \\ & =‖\left({\sigma }_{m,1}-r_{n,m_1}\right)\otimes {\sigma }_{m,2}+r_{n,m_1}⨂\left({\sigma }_{m,2}-r_{{n,m}_2}\right)‖\hfill \\ & \le {‖{\sigma }_{m,1}-r_{n,m_1}‖}_{L_{{\mu }_1\left(\mathbb{R}\right)}^1}·{‖{\sigma }_{m,2}‖}_{L_{{\mu }_2\left(\mathbb{R}\right)}^1}+{‖r_{n,m_1}‖}_{L_{{\mu }_1\left(\mathbb{R}\right)}^1}\hfill \\ & ·{‖{\sigma }_{m,2}-r_{n,m_2}‖}_{L_{{\mu }_2\left(\mathbb{R}\right)}^1}⟶0, n⟶\infty .\hfill \end{array}$$

(9)

We observe that the convergence to zero of the last term ${‖r_{n,m_1}‖}_{L_{{\mu }_1\left(\mathbb{R}\right)}^1}·{‖{\sigma }_{m,2}-r_{n,m_2}‖}_{L_{{\mu }_2\left(\mathbb{R}\right)}^1}$ is ensured by the boundedness of the sequence ${\left({‖r_{n,m_1}‖}_{L_{{\mu }_1\left(\mathbb{R}\right)}^1}\right)}_{n\ge 0}$, which converges to ${‖{\sigma }_{m,1}‖}_{L_{{\mu }_1\left(\mathbb{R}\right)}^1}<\mathrm{\infty }$ also using that ${‖{\sigma }_{m,2}-r_{n,m_2}‖}_{L_{{\mu }_2\left(\mathbb{R}\right)}^1}\to 0$ as $n\to \mathrm{\infty }.$ Thus, from Equations (6)–(9), we infer that the continuous compactly supported function $g$ taking nonnegative values at all points of ${\mathbb{R}}^2$ can be approximated in $L_{{\mu }_1\times {\mu }_2}^1\left({\mathbb{R}}^2\right)$ by sums of products $r_{n,m_1}\left(t_1\right)r_{n,m_2}\left(t_2\right)$ with nonnegative values $r_{n,m_i}\left(t_i\right)$ for all $t_i\in \mathbb{R},$ as $n⟶\mathrm{\infty }.$ Due to measure theory results, the same approximation-type result holds for any $\phi \in {\left(L_{{\mu }_1\times {\mu }_2}^1\left({\mathbb{R}}^2\right)\right)}_{+}$, since $\phi$ may be approximated by continuous compactly supported functions $g{\in \left(L_{{\mu }_1\times {\mu }_2}^1\left({\mathbb{R}}^2\right)\right)}_{+}$. Since such polynomial $r_{n,m_1}\otimes r_{n,m_2}$ and $r_{n,m_i}i=1,2$ is a sum of squares of polynomials in variable $t_i,$ we conclude that $\phi$ can be approximated by sums of products of the form $q_{n,1}^2\left(t_1\right)·q_{n,2}^2\left(t_2\right)=\left(q_{n,1}^2\otimes q_{n,2}^2\right)\left(t_1,t_2\right),q_{n,i}\in \mathbb{R}\left[t_i\right],i=1,2.$ This ends the proof. □

Let $S_i\subset \left[0,+\infty \right) i=1,2$ be compact nonempty subsets and let $S≔S_1\times S_2$ be their Cartesian product. By $C\left(S_1\right)⨂C\left(S_2\right)$, we denote the vector subspace of $C\left(S_1\times S_2\right)$ generated by all functions ${\phi }_1⨂{\phi }_2, {\phi }_i\in C\left(S_i\right), i=1,2.$

Theorem 2.

Any function $\phi \in {\left(C\left(S\right)\right)}_{+}$ is approximated in $C\left(S\right)$ by the restrictions to $S$ of polynomials of the form

$$r_1⨂r_2,\hspace{1em}{r}_i\in \mathbb{R}\left[t_i\right],\hspace{1em}{r}_i\left(t_i\right)\ge 0\hspace{1em}\forall t_i\in \left[0,+\infty \right), i=1,2.$$

(10)

It follows that any function $\phi \in {\left(C\left(S\right)\right)}_{+}$ is approximated in $C\left(S\right)$ by the restrictions to $S$ of sums of polynomials of the form

$$t_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right)=t_1^{s_1}{t}_2^{s_2}\left(q_1^2⨂q_2^2\right)\left(t_1,t_2\right), q_i\in \mathbb{R}\left[t_i\right],\hspace{1em}{s}_i\in \left\{0,1\right\}, i=1,2.$$

(11)

Proof. 

The proof follows the ideas like those from the proof of Theorem 1 except the fact that in case of the present theorem, any function ${\phi }_i\in C\left(S_i\right)$ can be approximated in this space by the restrictions to $S_i$ of polynomials that take nonnegative values at all points of the interval $\left[0,+\infty \right)$ (see Lemma 2 from [30]). As is well known, any such polynomial in variable $t_i$ is a finite sum of polynomials of the form (see [8], Theorem 3.12, p. 65)

$$t_i^{s_i}{q}_i^2\left(t_i\right), {\hspace{1em}q}_i\in \mathbb{R}\left[t_i\right],\hspace{1em}{s}_i\in \left\{0,1\right\},\hspace{1em}i=1,2 .$$

It results that any $\phi \in {\left(C\left(S\right)\right)}_{+}$ can be approximated by restriction to $S$ of sums of products having the following expressions

$$t_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right), q_i\in \mathbb{R}\left[t_i\right], s_i\in \left\{0,1\right\}, i=1,2.$$

This concludes the proof. □

3.2. Applications to the Moment Problem

Here is an application of Theorem 2 to the bidimensional moment problem on a Cartesian product of two compact subsets of $\left[0,+\infty \right).$

Theorem 3.

Let $S=S_1\times S_2$ be as mentioned before the statement of Theorem 2 and let $\mu$ be a positive regular Borel measure on $S.$ Let ${\left(y_{k,l}\right)}_{\left(k,l\right)\in {\mathbb{N}}^2}$ be a double-indexed sequence of real numbers. The following statements are equivalent.

(a)

There exists a unique positive regular Borel measure $\lambda$ on $S$ such that

$${\iint }_S{t}_1^{k_1}{t}_2^{l_2}d\lambda =y_{k_1,l_2}\mathrm{for} \mathrm{all} \left(k_1,l_2\right)\in {\mathbb{N}}^2,$$

(12)

$$\mathsf{\lambda }\left(B\right)\le \mu \left(B\right)\mathrm{for} \mathrm{all} \mathrm{Borel} \mathrm{subsets} B\subseteq S.$$

(13)

(b)

For $s_i\in \left\{0,1\right\}, i=1,2,$ any finite subsets $J_i\subset \mathbb{N},i=1,2,$ and all subsets $\left\{{\alpha }_j;j\in J_1\right\}, \left\{{\beta }_l;l\in J_2\right\}$ of real numbers, the following inequalities hold true:

$$\begin{array}{cc}\hfill 0\le {\displaystyle \sum _{j_1,l_1\in J_1}}{\alpha }_{j_1}{\alpha }_{j_2}& ·\left({\displaystyle \sum _{l_2,l_2\in J_2}}{\beta }_{l_1}{\beta }_{l_2}{y}_{j_1+j_1+s_1,l_1+l_2+s_2}\right)\hfill \\ & \le {\displaystyle \sum _{i_1,j_1\in J_1}}{\alpha }_{j_1}{\alpha }_{j_2}·\left({\displaystyle \sum _{\left(l_1,l_2\right)\in J_1\times J_2}}{\beta }_{l_1}{\beta }_{l_2}·{\iint }_S{t}_1^{i_1+j_1+s_1}·t_2^{i_2+j_2+s_2}d\mu \right)\hfill \end{array}$$

(14)

Proof. 

Let $T_0$ be defined on $\mathbb{R}\left[t_1,t_2\right]$ such that $T_0\left(t_1^{k_1}{t}_2^{l_2}\right)=y_{k_1,l_2}$ for all $\left(k_1,l_2\right)\in {\mathbb{N}}^2.$ Assuming (a) is known, we observe that inequalities (14) say that

$$\begin{array}{c}0\le T_0{(t}_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right))\le {\iint }_S\left({(t}_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right)\right)d\mu ,q_i\in \mathbb{R}\left[t_i\right],\\ s_i\in \left\{0,1\right\},i=1,2.\end{array}$$

(15)

Inequalities (14) follow directly from (12) and (13), since ${(t}_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right)$ take only nonnegative values. Hence, they define functions from ${\left(C\left(S\right)\right)}_{+}.$ Conversely, if the assertion (b) holds, then inequalities (13) and (17) (see below) hold as well. Indeed, summing term by term in inequalities with the same sense, the following further information follows: if $S_m\left(t_1,t_2\right)$ is a sum of polynomials like

$$\begin{array}{c}{\mathrm{t}}_1^{{\mathrm{s}}_1}{\mathrm{q}}_{1,\mathrm{m}}^2\left({\mathrm{t}}_1\right)·t_2^{s_2}{q}_{2,m}^2\left(t_2\right), \mathrm{we} \mathrm{infer} \mathrm{that}\\ 0\le T_0\left(S_m\right)\le {\iint }_S{S}_m\left(t_1,t_2\right)d\mu \forall m\in \mathbb{N}.\end{array}$$

(16)

Actually, as discussed in [8], any polynomial $r$ taking nonnegative values at all points $t\ge 0$ is a finite linear combination with nonnegative coefficients of special polynomials of the form $t^s{q}^2\left(t\right)=\sum _{i,j=0}^n{a}_i{a}_j{t}^{i+j+s},$ where $q$ is an arbitrary polynomial with real coefficients and $s\in \left\{0,1\right\}.$ The inequalities (14) stated at point (b) say that (15) holds true. Summing up a finite number of such inequalities, (16) holds as well. On the other hand, from Theorem 2, we already know that any $g\in {\left(C\left(S\right)\right)}_{+}$ is the limit of such sums $S_m$ in the Banach lattice $C\left(S\right)$, as $m\to \infty$ (see [30]). On the other hand, if ${\phi }_m\in {\left(\mathbb{R}\left[t_1{,t}_2\right]\right)}_{+},$ ${\phi }_m⟶0,$ then summing the terms appearing in Equation (11), it follows that we can write the next conclusion:

$$0\le T_0\left({\phi }_m\left(t_1,t_2\right)\right)\le {\iint }_S{\phi }_m\left(t_1,t_2\right)d\mu ⟶0,m⟶\mathrm{\infty }.$$

(17)

Thus, we are led to the next conclusion. Being given $g\in {\left(C\left(S\right)\right)}_{+},$ and $S_m$ a sum of special nonnegative polynomials appearing in (11), assume that $S_m-g\searrow 0$ in ${\left(C\left(S\right)\right)}_{+}.$ Then,

$$0\le S_m-g\le h_m≔{‖S_m-g‖}_{C\left(S\right)}·\mathbf{1}\in {\left(\mathbb{R}[{\mathrm{t}}_1,{\mathrm{t}}_2]\right)}_{+}\mathrm{for} \mathrm{all} m\in \mathbb{N}.$$

(18)

These further yields

$$T_0\left(S_m\right)\le {\iint }_S{S}_m\left(t_1,t_2\right)d\mu ={\iint }_S\left(S_m-g\right)d\mu +{\iint }_{S}gd\mu ⟶{\iint }_{S}gd\mu ,m\to \mathrm{\infty }.$$

(19)

For any $\epsilon >0,$ there exists $N\left(\epsilon \right)\in \mathbb{N}$ with $0\le S_m-g\le {‖S_m-g‖}_{C\left(S\right)}·\mathbf{1} \forall m\ge N\left(\epsilon \right).$ Assume that there is another sequence ${\left({\phi }_m\right)}_m$ of functions from the cone generated by all functions $t_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right)$, with ${\phi }_m\ge g\ge 0$ for all $m$, and ${\phi }_m\to g.$ Then, the inequalities

$$0{\le \phi }_m-g\le {\epsilon }_m·\mathbf{1},0{\le S}_m-g\le {\epsilon }_m·\mathbf{1} \mathrm{for} \mathrm{large} m\in \mathbb{N}, \mathrm{lead} \mathrm{to}:$$

$$-{\epsilon }_m·\mathbf{1}\le -\left({\phi }_m-g\right)\le \left(S_m-g\right)-\left({\phi }_m-g\right)\le S_m-g\le {\epsilon }_m·\mathbf{1}.$$

This yields

$${-{\epsilon }_m·\mathbf{1}\le S}_m-{\phi }_m=\left(S_m-g\right)-({\phi }_m-g)\le {\epsilon }_m·\mathbf{1}.$$

This further leads to

$$T_0\left(S_m-{\phi }_m\right)\le {\epsilon }_m{\iint }_S\mathbf{1}d\mu , T_0\left({\phi }_m-S_m\right)\le {\epsilon }_m{\iint }_S\mathbf{1}d\mu ,$$

$$\left|T_0\left(S_m-{\phi }_m\right)\right|\le {\epsilon }_m{\iint }_S\mathbf{1}d\mu ={\epsilon }_m·\mu \left(S\right)\to 0,m\to \mathrm{\infty }$$

(20)

Thus, denoting by $W$ the convex cone generated by all special positive polynomials

$${(t}_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right),$$

also using Theorem 2, for any $g\in {C\left(S\right)}_{+},$ there exists

$$\tilde{T}\left(g\right)≔\underset{\begin{array}{c}\phi \in W\\ \phi -g\searrow 0\end{array}}{\mathit{lim}}{T}_0\left(\phi \right)\le \underset{\begin{array}{c}\phi \in W\\ \phi -g\searrow 0\end{array}}{\mathit{lim}}{\iint }_S\phi d\mu ={\iint }_{S}gd\mu ,\tilde{T}\left(g\right)\ge 0,g\in {C\left(S\right)}_{+}.$$

(21)

Due to the additivity property of $T_0$ on $W,$ the additivity property of the extension $T$ of $T_0$ to ${C\left(S\right)}_{+}$ follows as well. For an arbitrary $s\in C\left(S\right)$, we define $\tilde{T}\left(s\right)≔\tilde{T}\left(s^{+}\right)-\tilde{T}\left(s^{-}\right).$ The linearity of $T_0$ on $W$ ensures the positively homogeneity property for $T$ on $W$ and the linearity of $T$ on $Y≔W-W.$ Then, the linearity of the extension $\tilde{T}$ to $C\left(S\right)=$ ${C\left(S\right)}_{+}-{C\left(S\right)}_{+}$ follows as well. The continuity of $\tilde{T}$ at zero and then on the entire space $C\left(S\right)$ follows from its positivity given by (16) via its linearity (see [4] and, for a more general result regarding the continuity of any positive linear operator on ordered Banach spaces, see [22], Lemma 2, p. 2). The positive linear functional $\tilde{T}$ is represented by a positive regular Borel measure $\lambda$ defined on the sigma algebra $\mathfrak{B}\left(S\right)$ of all Borel subsets of $S.$ Namely, the polynomials $t_1^{k_1}{t}_2^{l_2}$ are special polynomials from the class of polynomials $t_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right)$ and ${\iint }_S{t}_1^{k_1}{t}_2^{l_2}d\lambda =T_0\left(t_1^{k_1}{t}_2^{l_2}\right)≔y_{k_1,l_2}.$ The positivity of the measure $\lambda$ is a consequence of the positiveness property of representing positive linear functionals on $K$ by positive regular Borel measures on $S,$ as the Riesz representation theorem [2] states. To prove the last inequality (13) of the point (a), let $B$ be a Borel subset in $K$ with $\mu \left(B\right)>0.$ Then, ${\chi }_B\in {\left(L_{\mu }^1\left(S\right)\right)}_{+}$ can be approximated by a sequence $S_m, m\in \mathbb{N}$, of polynomial functions from the cone $W$ generated by the polynomials having the special form

$$t_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right),q_i\in \mathbb{R}\left[t_i\right],s_i\in \left\{0,1\right\},i=1,2.$$

For each such special polynomial, the second inequality (14) says that

$${\iint }_S{t}_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right)d\lambda =T_0\left(t_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right)\right)\le {\iint }_S{t}_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right)d\mu .$$

Summing these inequalities written for each term of the sum $S_m,$ then passing to the limit as $m⟶\infty ,$ we find that the following conclusion holds:

$$\lambda \left(B\right)={\iint }_S{\chi }_{B}d\lambda =\underset{m}{\mathit{lim}}{\iint }_S{S}_{m}d\lambda \le \underset{m}{\mathit{lim}}{\iint }_S{S}_{m}d\mu ={\iint }_S{\chi }_{B}d\mu =\mu \left(B\right).$$

The conclusion is that $\lambda \left(B\right)\le \mu \left(B\right)$ for all Borel subsets of $S.$ Hence, (13) holds true. This ends the proof. □

Corollary 1.

With the notations and under the hypothesis of Theorem 3, we infer that the solution measure $\lambda$ has the following representation:

$$d\lambda =\psi d\mu ,\mathrm{where} \psi \in L_{\mu }^{\mathrm{\infty }}\left(S\right),0\le \psi \le \mathbf{1} \mathrm{in} L_{\mu }^{\mathrm{\infty }}\left(S\right).$$

Proof. 

One applies standard-type measure theory results [1,2]. This ends the proof. □

Using Theorems 2 and 3 proved above, by a similar proof to that of Theorem 3, the following more general result holds true.

Theorem 4.

Let $S=S_1\times S_2$ be as in Theorems 2 and 3 and ${\left(y_{k,l}\right)}_{\left(k,l\right)\in {\mathbb{N}}^2}$ be a sequence of functions from a Banach lattice $Y$. Assume that $Y$ is also a Banach algebra over the real field, such that $y^2\in Y_{+}$ for all $y\in Y,$ and $y_1{·y}_2\in Y_{+}$ for all $y_1,y_2\in Y_{+}.$ Also, assume that $Y$ has an order unit $e,$ which is also the unit element for the multiplication operation in $Y.$ Let  ${\left(p_{k,l}\right)}_{\left(k,l\right)\in {\mathbb{N}}^2}$ be the sequence of polynomials defined by

$$p_{k,l}\left(t_1,t_2\right)≔t_1^{k_1}{t}_2^{l_2},\left(t_1,t_2\right)\in S, \left(k_1,l_2\right)\in {\mathbb{N}}^2,$$

(22)

 and $T_2$ a positive linear operator mapping $C\left(S\right)$ into $Y$. The following statements are equivalent:

(a)

There exists a unique linear operator $T$ mapping $C\left(S\right)$ into $Y$, such that

$$T\left(p_{k,l}\right)=y_{k,l}, \left(k,l\right)\in {\mathbb{N}}^2;$$

(23)

$$0\le T\left(\phi \right)\le T_2\left(\phi \right), \phi \in {\left(C\left(S\right)\right)}_{+}.$$

(24)

(b)

For $s_i\in \left\{0,1\right\}, i=1,2,$ any finite subsets $J_i\subset \mathbb{N},i=1,2,$ and all subsets $\left\{{\alpha }_j;j\in J_1\right\}, \left\{{\beta }_l;l\in J_2\right\}$ of real numbers, the following inequalities hold true:

$$\begin{array}{c}0\le {\displaystyle \sum _{i_1,j_1\in J_1}}{\alpha }_{i_1}{\alpha }_{j_1}\left({\displaystyle \sum _{i_2{,j}_2\in J_2}}{\beta }_{i_2}{\beta }_{j_2}{y}_{i_1+j_{1+}{s}_1,i_2+j_{2+}{s}_2}\right)\le \\ \sum _{j_1,l_1\in J_1}{\alpha }_{j_1}{\alpha }_{j_2}\left(\sum _{i_{2, }{j}_2\in J_2}{\beta }_{i_2}{\beta }_{j_2}{T}_2\left(p_{i_1+j_{1+}{s}_1,i_2+j_{2+}{s}_2}\right)\right).\end{array}$$

Proof. 

One applies the method of the proof of Theorem 3, where the linear functional $T_2$ defined there by the measure $d\mu$ now is replaced by an arbitrary bounded linear operator $T_2:C\left(S\right)⟶Y.$ According to Equation (10), condition (b) is equivalent to

$$0\le T_0{(t}_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right))\le T_2{(t}_1^{s_1}{q}_1^2\left(t_1\right)·t_2^{s_2}{q}_2^2\left(t_2\right)),q_i\in \mathbb{R}\left[t_i\right],s_i\in \left\{0,1\right\},t_i\in [0,+\mathrm{\infty }),i=1,2.$$

Summing, and applying the method from the proof of Theorem 3, Equation (16), and the notations there with the change in upper constraint defined by $T_2$ instead of $d\mu ,$ one obtains

$$0\le \tilde{T}\left(g\right)≔\underset{\begin{array}{c}\phi \in W\\ \phi -g\searrow 0\end{array}}{\mathit{lim}}{T}_0\left(\phi \right)\le T_2\left(g\right),\forall g\in {\left(C\left(S\right)\right)}_{+}.$$

The rest of the proof follows like that of Theorem 3. This ends the proof. □

Here is a short list of examples of spaces $Y$ for which Theorem 4 holds, taking $Y$ as codomain of the operators $T_2$ and $T.$

Example 1.

(i) $Y_1=C\left(S\right)$. Here, $e=\mathbf{1}.$ (ii) Let ${ Y}_2$ be the commutative algebra (which is also an order complete Banach lattice), constructed in [5], pp. 303–305, consisting of special self-adjoint operators acting on a Hilbert space $H$ and satisfying a commutativity-type condition. In this case, $e=I$ is the identity operator $I:H\to H$ (see Section 2 above, point 5). (iii) Let $Y_3=L_{\mu }^{\infty }\left(\mathbb{R}\right);$ then $e=\widehat{\mathbf{1}}$. (iv) If $Y_4=l^{\infty }\left(\mathbb{N}\right), e=\left(\mathrm{1,1},\dots ,1,\dots \right)$, the sequence having all its components equal to the number $1,$ then $e$ verifies the required conditions.

Theorem 5.

Let $H$ be a Hilbert space, $A_1,A_2$ two commuting positive self-adjoint operators mapping $H$ into $H,$ and let $Y≔Y\left(A_1,A_2\right)$ be the commutative Banach algebra of self-adjoint operators defined below in Equation (25). If we denote by $S_i$ the spectrum of the positive operator $A_i, i=1,2,$ $S≔S_1\times S_2,$ let us consider the positive linear operator ${\tilde{T}}_2$ mapping $C\left(S_1\times S_2\right)$ into $Y\left(A_1,A_2\right)$ that verifies the equalities ${\tilde{T}}_2\left(t_1^{j_1}{t}_2^{j_2}\right)=A_1^{j_1}{A}_2^{j_2}$ for all $\left(j_1,j_2\right)\in {\mathbb{N}}^2.$ Being given a sequence ${\left(U_{\left(j_1,j_2\right)}\right)}_{\left(j_1,j_2\right)\in {\mathbb{N}}^2}$ of elements from $Y\left(A_1,A_2\right),$ the following statements are equivalent:

(a)

There exists a unique linear operator  $T$ mapping $C\left(S\right)$ into $Y$, such that

$$\mathrm{T}\left(t_1^{j_1}{t}_2^{j_2}\right)=U_{\left(j_1,j_2\right)}, \left(j_1,j_2\right)\in {\mathbb{N}}^2,$$

$$0\le T\left(\phi \right)\le {\tilde{T}}_2\left(\phi \right), \phi \in {\left(C\left(S\right)\right)}_{+}.$$

(b)

For $s_i\in \left\{0,1\right\}, i=1,2,$ any finite subsets $J_i\subset \mathbb{N},i=1,2,$ and all subsets $\left\{{\alpha }_{j_1};j_1\in J_1\right\}, \left\{{\beta }_{j_2};j_2\in J_2\right\}$ of real numbers, the following inequalities hold true:

$$\begin{array}{c}0\le {\displaystyle \sum _{i_1,j_1\in J_1}}{\alpha }_{i_1}{\alpha }_{j_1}\left({\displaystyle \sum _{i_2{,j}_2\in J_2}}{\beta }_{i_2}{\beta }_{j_2}{U}_{i_1+j_{1+}{s}_1,i_2+j_{2+}{s}_2}\right)\le \\ \sum _{j_1,l_1\in J_1}{\alpha }_{j_1}{\alpha }_{j_2}\left(\sum _{i_{2,}{j}_2\in J_2}{\beta }_{i_2}{\beta }_{j_2}{A}_1^{i_1+j_{1+}{s}_1}{A}_2^{i_2+j_{2+}{s}_2}\right).\end{array}$$

Proof. 

The proof follows from Theorem 4, where $Y$ stands for $Y\left(A_1, A_2\right)$ defined below (see (25)) and $y_{k,l}$ stands for $U_{k,l}$, $\left(k,l\right)\in {\mathbb{N}}^2,$ and $T_2$ stands for ${\tilde{T}}_2,$ also using a part of the proof of Theorem 3. Namely, to prove the present theorem, we apply Theorem 4 to

$$S_i≔Spectr\left(A_i\right),i=1,2,$$

$$T_2\left(t_1^{j_1}{t}_2^{j_2}\right)≔A_1^{j_1}{A}_2^{j_2}\mathrm{for} \mathrm{all} \left(j_1,j_2\right)\in {\mathbb{N}}^2.$$

Then, we extend $T_2$ by linearity to $\mathbb{R}[t_1,t_2]$. For a fixed function ${\phi }_i\in C\left(S_i\right),$ we denote by ${\phi }_i\left(A_i\right)\in Y\left(A_i\right)$ the self-adjoint operator obtained as an image of ${\phi }_i$ through the linear isometry ${\Psi }_i$ from $C\left(S_i\right)$ into $Y\left(A_i\right),$ which was naturally obtained by means of the functional calculus for continuous real functions on the spectrum of the self-adjoint operator $A_i, i=1,2.$ Next, we define the Banach algebra $Y\left(A_1,A_2\right),$ as follows. If $A_1, A_2$ are commuting self-adjoint operators on $H,$ denoting $\mathcal{S}\mathcal{A}\left(H\right)$ the ordered Banach space of all self-adjoint operators acting on $H,$ one defines

$$Y\left(A_1,A_2\right)≔cl\left(Span\left\{A_1^{j_1}{A}_2^{j_2}; \left(j_1,j_2\right)\in {\mathbb{N}}^2\right\}\right).$$

(25)

In Equation (25), the closure appearing on the right-hand side is the topological closure of the vector subspace $Span\left\{A_1^{j_1}{A}_2^{j_2}; \left(j_1,j_2\right)\in {\mathbb{N}}^2\right\}$ in the ordered Banach space $\mathcal{S}\mathcal{A}\left(H\right).$ Then, $Y\left(A_1,A_2\right)$ is a commutative algebra of self-adjoint operators, over the real field, and an ordered Banach space. Moreover, each operator $B\in Y\left(A_1, A_2\right)$ is self-adjoint (as a limit of a sequence of self-adjoint operators, in the space $\mathcal{S}\mathcal{A}(H$)) and $B$ commutes with any other self-adjoint operator $U$ commuting with both operators $A_1, A_2.$ Next, for polynomials $q_i\in \mathbb{R}\left[t_i\right], i=1,2$, we define

$${ T}_{1,2}\left(q_1⨂q_2\right)≔q_1\left(A_1\right)q_2\left(A_2\right)\in Y\left(A_1, A_2\right),\hspace{1em}\hspace{1em}{q}_i\in \mathbb{R}\left[t_i\right], i=1,2.$$

(26)

Then, for ${\phi }_1\in C\left(S_1\right),$ and the sequence of polynomials $q_{1,m}⟶{\phi }_1$ in $C\left(S_1\right),$ passing to the limit as $m⟶\infty ,$ we obtain a bilinear operator $T_2,$

$$T_2:C\left(S_1\right)\times \mathbb{R}\left[t_2\right]⟶ Y\left(A_1, A_2\right), T_2\left(f,q\right)≔f\left(A_1\right)q\left(A_2\right).$$

(27)

On the other hand, returning to the space $C\left(S_2\right)$ and the isometry ${\Psi }_2$, we infer the following facts. For any fixed $g\in C\left(S_2\right),$ and for an arbitrary sequence ${\left(q_{2,m}\right)}_{m\ge 0}$ of polynomials

$$q_{2,m}\in \mathbb{R}\left[t_2\right]{, with q}_{2,m}⟶g, m⟶\infty$$

since ${\tilde{T}}_2\left(1,q_{2,m}\right)=q_{2,m}\left(A_2\right)={\Psi }_2\left(q_{2,m}\right)$ for all $m,$ we obtain

$${\Psi }_2\left(g\right)=\underset{m}{\mathit{lim}}{\Psi }_2\left(q_{2,m}\right)=\underset{m}{\mathit{lim}}{q}_{2,m}\left(A_2\right).$$

Hence, passing to the limit in Equation (27) for fixed $f$ and arbitrary $q_{2,m}⟶g,$ we conclude that

$$T_2\left(f,g\right)≔\underset{m}{lim}f\left(A_1\right)q_{2,m}\left(A_2\right)=f\left(A_1\right){\Psi }_2\left(g\right)=f\left(A_1\right)g\left(A_2\right).$$

(28)

In other words, (27) stays valid for any $\left(f,g\right)\in C\left(S_1\right)\times C\left(S_2\right).$ Hence,

$${\tilde{T}}_2\left(f,g\right)={\Psi }_1\left(f\right){\Psi }_2\left(g\right),\left(f,g\right)\in C\left(S_1\right)\times C\left(S_2\right).$$

(29)

Then, we extend $T_{1,2}$ defined by (26) to a linear operator on

$$X_{1,2}≔Span\left\{f⨂g;f\in C\left(S_1\right),g\in C\left(S_2\right)\right\}\subset C\left(S_1\times S_2\right),$$

(30)

in the obvious way, the codomain being $Y\left(A_1, A_2\right)$ as follows:

$$T_{1,2}\left({\sum }_{i=0}^p{{\alpha }_{i}f}_i⨂g_i\right)≔{\sum }_{i=0}^p{a}_i{\Psi }_1\left(f_i\right){\Psi }_2\left(g_i\right).$$

(31)

We observe that to simplify the notation, $T_{1,2}$ has been not changed when passing from the generators of the subspace $C\left(S_1\right)⨂C\left(S_2\right)$ to the generated subspace $C\left(S_1\right)⨂C\left(S_2\right)$. To finish the proof, we must show that $T_{1,2}$ maps the positive cone ${\left(C\left(S\right)\right)}_{+}$ into ${cl\left(Y\left(A_1, A_2\right)\right)}_{+}.$ Let $\phi \in {\left(C\left(S_1\times S_2\right)\right)}_{+}.$ According to the Stone–Weierstrass theorem, also using Theorem 2, Equation (11), there exist approximating nonnegative polynomials on $\left[0,+\infty \right),$ as discussed in Equation (10) (see Theorem 2 above). Due to the explicit expression of such a polynomial (see (11)), and using the properties of the isomorphisms ${\Psi }_i, i=1,2,$ we observe that

$${\Psi }_l\left(t_l^{s_l}{q}_l^2\left(t_l\right)\right)=t_l^{s_l}{q}_l^2\left(t_l\right)=A_l^{s_l}{q}_l^2\left(A_l\right)\in (Y{\left(A_1, A_2\right))}_{+}forall{t}_l\in \left[0,+\infty \right),$$

(32)

$s_l\in \left\{0,1\right\}, q_l\in \mathbb{R}\left[t_l\right], l=1,2.$ This is true, since the square of any self-adjoint operator is self-adjoint and positive, and the product of commuting self-adjoint positive operators is self-adjoint and positive. Hence, for nonnegative numbers $a_i,$ the right-hand side of (31) is an element from ${\left(Y\left(A_1, A_2\right)\right)}_{+}.$ Thus,

$$T_{1,2}\left(r_1⨂r_2\right)\ge 0\mathrm{in} Y\left(A_1, A_2\right)\mathrm{for} \mathrm{all} r_l\in {\left(\mathbb{R}\left[0,+\infty \right)\right)}_{+}, l=1,2.$$

(33)

From the continuity of the operators ${\mathsf{\Psi }}_{\mathrm{i}},\mathrm{i}=1,2,$ we infer the continuity of ${\mathrm{T}}_{1,2}$ appearing in (31) on the dense subspace ${\mathrm{X}}_{1,2}$ defined in Equation (30). As discussed in [4, pp. 228–229], since the positive cone ${\left(C\left(S_1\times S_2\right)\right)}_{+}$ has interior points (such as the constant function $\mathbf{1}$), from the positivity claimed by (33), we derive the continuity of $T_{1,2}$ appearing in (31) on the dense subspace $X_{1,2}$ of $C\left(S_1\times S_2\right).$ Hence, $T_{1,2}$ can be extended by linearity and continuity to a linear continuous (and multiplicative) operator ${\tilde{T}}_2$ as follows:

$${\tilde{T}}_2:C\left(S_1\times S_2\right)⟶Y\left(A_1, A_2\right).$$

If $\phi \in {\left(C\left(S_1\times S_2\right)\right)}_{+}$ and if $\mathsf{\phi }=\underset{\mathrm{p}}{\lim }{\mathrm{r}}_{\mathrm{p}},{\mathrm{r}}_{\mathrm{p}}={\mathrm{r}}_{1,\mathrm{p}}⨂{\mathrm{r}}_{2,\mathrm{p},}$ ${\mathrm{r}}_{\mathrm{l},\mathrm{p}}\in {\left(\mathbb{R}\left[0,+\mathrm{\infty }\right)\right)}_{+},\mathrm{l}=1,2,$ then passing to the limit as $\mathrm{p}\to \mathrm{\infty }$ via the continuity of ${\tilde{\mathrm{T}}}_2$ and positivity of ${\mathrm{T}}_{1,2}$ on each ${\mathrm{r}}_{1,\mathrm{p}}⨂{\mathrm{r}}_{2,\mathrm{p}},\mathrm{p}\in \mathbb{N}$ (see Equation (33)). Let us observe that for each fixed $p\in \mathbb{N},$ ${T_2\left(r_p\right)=r}_{1,p}\left(A_1\right)·r_{2,p}\left(A_2\right)\ge 0$ in $Y\left(A_1, A_2\right)$ as a sum of products of permutable self-adjoint positive operators. Since the positive cone $Y{\left(A_1, A_2\right))}_{+}$ is topologically closed, we are led to the desired conclusion of this theorem as follows:

$${\tilde{T}}_2\left(\phi \right)≔\underset{p}{\mathit{lim}}{T}_2(r_p)=\underset{p}{lim}(r_{1,p}\left(A_1\right)·r_{2,p}\left(A_2\right))\in (Y{\left(A_1, A_2\right))}_{+}, \phi \in {\left(C\left(S_1\times S_2\right)\right)}_{+}.$$

Thus, ${\tilde{\mathrm{T}}}_2$ is also positive on the positive cone $(\mathrm{Y}{\left({\mathrm{A}}_1,{\mathrm{A}}_2\right))}_{+}.$ The rest of the conclusions follow from Theorem 4. □

Corollary 2.

With the notations and under the hypothesis of Theorem 5, if the sequence ${\left(U_{\left(j_1,j_2\right)}\right)}_{\left(j_1,j_2\right)\in {\mathbb{N}}^2}$ of self-adjoint operators satisfies conditions (b), and  $T$ is the solution of the constrained interpolation problem claimed at point (a), then

$$‖\tilde{T}‖\le 1, 0\le U_{\left(j_1,j_2\right)}\le A_1^{j_1}{A}_2^{j_2}, ‖U_{\left(j_1,j_2\right)}‖\le {‖A_1‖}^{j_1}·{‖A_2‖}^{j_2}.$$

Proof. 

Since $\tilde{T}$ is linear and positive between Banach lattices $C\left(S_1\times S_2\right)$ and $Y\left(A_1,A_2\right),$ we infer that $‖\tilde{T}‖=‖\tilde{T}\left(\mathbf{1}\right)‖\le ‖{\tilde{T}}_2\left(\mathbf{1}\right)‖=‖I‖=1.$ The other inequalities are obvious. □

Corollary 3.

The statement of Theorem 5 stays valid for $n\times n$ symmetric commuting matrices  $A_1, A_2, U_{\left(j_1,j_2\right)}$,  $\left(j_1,j_2\right)\in {\mathbb{N}}^2,$ with real entries, whose corresponding linear operators satisfy the hypothesis of Theorem 5.

Proof. 

We apply Theorem 5 to the $n$-dimensional real Hilbert space ${\mathbb{R}}^n$ and to the spectrum $S_i$ of the matrix $A_i, i=1,2$. The operatorial norm of such a matrix is the norm of the linear self-adjoint operator defined by that matrix. □

4. Discussion

We have proved results on the existence and uniqueness of the solution for the bidimensional moment problem in $L_{\mu }^1$ type spaces, where $\mu ={\mu }_1\times {\mu }_2$ is the product of two moment determinate measures ${\mu }_i$ on $\mathbb{R},i=1,2.$ This result is using the corresponding approximation theorem of any function from ${(L}_{\mu }^1{\left({\mathbb{R}}^2\right))}_{+}$ by special nonnegative $p=p_1⨂p_2$ polynomials on the entire plane ${\mathbb{R}}^2$ in the norm of the space $L_{\mu }^1\left({\mathbb{R}}^2\right)$ (Theorem 1). Here, each of the polynomials $p_i, i=1,2$ is nonnegative on $\mathbb{R}$ and hence is a sum of squares. A similar property is pointed out in Theorem 2. This allows us to use quadratic forms in characterizing the existence of the solution. All the results remain valid when passing from the case $n=2$ to the case of the $n$-dimensional space ${\mathbb{R}}^n, n\ge 3.$ Thus, the $n$-dimensional moment problem is solved in terms of quadratic forms. In Theorems 3–5 and whenever the same function spaces are involved as domains of the solutions, the uniqueness of the solution follows via its continuity and density of polynomials in the domain space. The next applied polynomial approximation result refers to nonnegative continuous functions defined on a compact subset $S=S_1\times S_2\subset {\mathbb{R}}_{+}^2.$ Each such function can be approximated in the space $C\left(S\right)$ by restrictions to $S$ of products $p_1⨂p_2,$ $p_i$ nonnegative polynomials on ${\mathbb{R}}_{+}, i=1,2.$ Hence, again, we could derive a solution to the moment problem in terms of products of quadratic forms. From here, we can see connections with numerical examples and linear algebra.

5. Conclusions

The present article provides characterizations for the existence and uniqueness of the full multidimensional moment problem in terms of quadratic forms. The scalar valued as well as the operator valued cases are under attention. We use proving the results by passing to the limit process from polynomials to functions. However, in our statements, the limit sign does not appear, although it is essentially used during the proofs. As a first direction for future work, we mention passing from $n=2$ to arbitrary dimension $n\ge 3.$ Considering other function spaces in $n$ dimensions, where a useful polynomial approximation result does work, could be an interesting subject. For example, on ${\mathbb{R}}^n,$ in the spaces mentioned in Theorem 1, any nonnegative (real valued) function is the limit of a sequence whose terms are the sums of squares of polynomials. However, there are spaces containing continuous compactly supported functions defined on unbounded closed subsets that are not semialgebraic sets. Although the polynomial approximation of nonnegative continuous compactly supported functions by nonnegative polynomials still hold on such subsets, the explicit form of these polynomials in terms of sums of squares is unknown.