Applications of the Hahn-Banach Theorem, a Solution of the Moment Problem and the Related Approximation

Abstract

We start by an application the of Krein–Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn–Banach-type theorems and polynomial approximation to obtain recent results on the moment problem on the unbounded closed interval $[0,+\infty )$. Necessary and sufficient conditions for the existence and uniqueness of the solution are pointed out. Operator-valued moment problems and a scalar-valued moment problem are solved.

1. Introduction

The aim of this paper is to review a few main results in analysis and functional analysis, whose common point consists of Hahn–Banach-type theorems and their consequences. The first part is devoted to geometric aspects in functional analysis, such as the Krein–Milman theorem. As is well-known, Bernstein integral representation of completely monotonic functions in terms of the exponential functions of negative exponents can be proved via the Krein–Milman theorem. The latter theorem is a consequence of the geometric form of the Hahn–Banach theorem. For this first part, the main sources of information can be found in references [1,2,3,4,5,6,7,8]. Namely, the book [1] contains basic facts in functional analysis, from simple to higher-level results. Duality theory and weak topologies and weak compactness and related results on weakly continuous functions are carefully presented. In the last part of the book (Chapter V), the main results on ordered topological vector spaces, including Banach lattices, are reviewed. Monograph [2] is strongly related to the first part of the present review paper, by means of the way of presenting the Bernstein integral representation theorem of completely monotonic functions. The book [3] contains general information on convexity and special modern-related topics. Among them, we mention convexity in spaces of matrices and majorization theory. The book [4] completes various topics on functions of one real variable, including elements of dynamical systems. The relationship with other topics is done with the aid of comments and exercises, which accompany each chapter. In the book [5], basic notions in real analysis and complex analysis are studied together. Each chapter builds upon the other. Challenging exercises complete each chapter. Measure theory and Jensen inequality are also discussed. The reference [6] represents an excellent method of understanding important topics in functional analysis. In the present review paper, we apply functional calculus for real-valued continuous functions on the spectrum of a self-adjoint operator as it is presented in [6]. In [7], the reader can find the general theory on ordered vector spaces and on topological ordered vector spaces, as well as properties of linear operators acting on such spaces. In the present article, we use the construction from [7] regarding the codomain of our operator-valued solutions of moment problems. This codomain space is an order-complete Banach lattice and a commutative algebra whose elements are self-adjoint operators acting on a Hilbert space. Therefore, in using the Hahn–Banach theorem for the extension of linear operators, having this space as codomain can be applied. In reference [8], the authors present elements of Choquet theory, abstract and applied potential theory, and special convex cones of functions. In [9], aspects of the uniform approximation of continuous functions are discussed. First, Hahn–Banach-type results stated in terms of subdifferentiability of convex operators were published in [10,11]. The references [10,11,12] continue and complete the studies from [1,2,3,4,5,6,7,8]. In the article [13], the K-moment problem for a closed (not necessarily bounded) basic semi-algebraic set K of ${\mathbb{R}}^n$ is completely solved. The third part of this study is devoted to the moment problem. The existence part of this problem is a constrained-extension-type result of a linear functional or linear operator, from the subspace of polynomials to a larger space of functions, preserving the positivity and that of being dominated by a given convex operator, on the positive cone of the domain space. The present review paper provides necessary and sufficient conditions for the existence and uniqueness of the solution for the full moment problem on compact or on the entire interval $\left[0,+\infty \right).$ In its classical formulation, being given a sequence ${\left(m_n\right)}_{n\ge 0}$ of real numbers, such a solution is a positive regular Borel measure $\mu$ on $\left[0,+\infty \right),$ with prescribed moments $m_n={\int }_0^{+\infty }{t}^{n}d\mu \left(t\right)$ of all orders $n\in \mathbb{N}≔\left\{\mathrm{0,1},2,\dots \right\}.$ In some cases, an upper boundedness condition on the solution, such as the following, is also required.

$${\int }_0^{+\infty }\phi \left(t\right)d\mu \left(t\right)\le P\left(\phi \right),\hspace{1em}\forall \phi \in {\left(L_{\mu }^1\left(\left[0,+\infty \right)\right)\right)}_{+},$$

(1)

Here, $P:{\left(L_{\mu }^1\left(\left[0,+\infty \right)\right)\right)}_{+}⟶{\mathbb{R}}_{+}$ is a given convex functional. If a solution does exist and if it is unique, the problem of its construction might be of interest. Since the values of the solution on the subspace $\mathbb{R}\left[t\right]$ of polynomials are known, the existence of the solution consists of applying an extension-type result of the linear functional to a positive linear functional $T$ defined on $L_{\mu }^1\left(\left[0,+\infty \right)\right)$(here $J_0\subset \mathbb{N}$ is an arbitrary finite subset), from $\mathbb{R}\left[t\right]$ to $L_{\mu }^1\left(\left[0,+\infty \right)\right)$ or to another Banach lattice containing $\mathbb{R}\left[t\right]$.

$$T_0:\mathbb{R}\left[t\right]⟶\mathbb{R},\hspace{1em}{T}_0\left(\sum _{j\in J_0}{\alpha }_j{p}_j\right)≔\sum _{j\in J_0}{\alpha }_j{m}_j, {\alpha }_j\in \mathbb{R},\hspace{1em}{p}_j\left(t\right)≔t^j, j\in \mathbb{N},$$

(2)

In the article [13], the K-moment problem for a closed (not necessarily bounded) basic semi-algebraic set K of ${\mathbb{R}}^n$ is completely solved. Referring to the moment problem, we are using related results and paragraphs from the book/chapters of [14], the article [15], and the articles or review papers [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. The ordering of the references follows approximately the chronological date of their publication. In [15], differences between the multidimensional and the one-dimensional case are pointed out (see Lemma 1 and Theorem 4 there). Among other results, the author of article [16] applies convexity results to the moment problem for a large class of compact subsets of ${\mathbb{R}}^n, n\ge 2$. The form of positive polynomials on such a compact subset $K$, in terms of the specific polynomials defining the set $K,$ is also determined. More precise results hold true when $K$ is defined by a means of finite number polynomials; that is, the set $K$ is a semi-algebraic compact. This means that the compact $K$ is defined as follows:

$$K=\left\{x\in {\mathbb{R}}^n; r_1\left(x\right)\ge 0,\dots ,r_k\left(x\right)\ge 0\right\}, k\in \mathbb{N}, k\ge 1.$$

These results were first published in [17]. In [19], applications and completions of the author’s results to those from [16,17], respectively, are stated. The articles [20,21] continue the study from [17] for a class of special polynomials $r_i$, defining the set compact $K.$ Consequently, more specific information is contained in the conclusions. The decomposition of any nonnegative polynomial on such a compact as a product in terms of the given polynomials $r_i$, $i=1,\dots ,k$ requires a special attention. Unlike the case $n=1,$ for $n\ge 2,$ there exists nonnegative polynomials on ${\mathbb{R}}^n$ which are not sums of squares (see [15], Lemma 1). In addition, there are positive definite sequences which are not moment sequences (see [15], Theorem 4). However, the articles [16,17,19,20,21] solve these problems when instead of ${\mathbb{R}}^n$ we consider special compact subsets of ${\mathbb{R}}^n$, $n\ge 2]$. In [18], interesting connections between the moment problem and operator theory are pointed out. The article [19] applies a generalized Hahn–Banach-type result to the moment problem, with two constraints on the solution. In the general case of the multidimensional moment problem, $S$ is a bounded or unbounded closed subset of ${\mathbb{R}}^n, n\ge 2,$ $j=\left(j_1,\dots ,j_n\right)\in {\mathbb{N}}^n$ is a multi-index, $p_j\left(t\right)=t^j≔t_1^{j_1}\cdots t_n^{j_n},t=\left(t_1,\dots ,t_n\right)\in S, X$ is a Banach lattice of functions containing $\mathbb{R}\left[t_1,\dots ,t_n\right]$ and $C_c\left(S\right),$ $Y$ is an order-complete Banach lattice, ${\left(m_j\right)}_{j\in {\mathbb{N}}^n}$ is a given sequence in $Y,$ and $P:X_{+}⟶Y$ (or $P:X⟶Y$) a convex continuous operator. Usually, by a solution of the full $n-$ dimensional vector valued moment problem we mean characterizing the existence and uniqueness of a linear operator $T$ from $X$ into $Y,$ satisfying the following moment conditions (3) and constraints (4), below, on the positive cone $X_{+}$ of $X$.

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

(3)

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

(4)

We recall that in the case $Y=\mathbb{R},$ an (M)-determinate measure $\mu$ on $S$ is, by definition, uniquely determined by its moments $m_j$, defined by (3), and $T$ satisfies (4). In this case, there exists a positive regular representing Borel measure $\mu$ for the positive linear functional $T$ defined by the interpolation formulae (3) and the constraints (4) on it:$T\left(\phi \right)={\int }_S\phi d\mu , \phi \in X$. Thus, the conditions (3) and (4), respectively, should be written as follows:

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

and, respectively,

$$0\le {\int }_S\phi d\mu \le P\left(\phi \right), \forall \phi \in X_{+}.$$

Then, for the uniqueness of the solution, it is sufficient to prove the density of $\mathbb{R}\left[t_1,\dots ,t_n\right]$ in $X.$ The article [22] is devoted to new sufficient conditions for the determinacy of probability densities on $\mathbb{R}$ and on $\left[0,+\infty \right).$ In the article [23], referring to the reduced Stieltjes and Hamburger moment problems when the Maximum Entropy does not exist, the authors find methods of approximating the unknown but true underlying distribution. In the study [24], earlier and new results involving determinacy and indeterminacy in Hamburger and Stieltjes moment problems have been recently pointed out. These theorems are accompanied by geometric interpretations, new arguments for classical results, and new numerical illustrations. The articles [25,26] provide new results on operator theory and their relationship with the moment problem. Article [27] has no specified connection with the moment problem, but the continuity of linear positive operators is useful in solving moment problems. In addition, in the same paper, the characterization of the isotonicity property (monotone increasing property) of a convex operator on the positive cone of the domain space is discussed via a general Hahn–Banach-type result. On the other hand, in [28], the density of nonnegative polynomials in $X_{+}={\left(L_{\mu }^1\left(\left[0,+\infty \right)\right)\right)}_{+}$ was proved for any positive regular (M)-determinate measure $\mu ,$ with finite moments of all orders. Consequently, for such measures $\mu$ on $\left[0,+\infty \right)$, $\mathbb{R}\left[t\right]$ is dense in $L_{\mu }^1\left(\left[0,+\infty \right)\right)$. This result also works for any unbounded closed subset $S\subseteq {\mathbb{R}}^n,$ endowed with a positive regular moment determinate Borel measure $\mu ,$ with finite moments of all orders. Now let us notice that the first condition (4) implies the continuity of the solution $T$ on $X$ and its uniqueness follows via the density of polynomials in $L_{\mu }^1\left(\left[0,+\infty \right)\right).$ Indeed, any positive linear operator acting on Banach lattices is continuous (see [27] for the more general property of linear positive operators acting on ordered Banach spaces). In [28], polynomial approximation on closed unbounded subsets was studied. These results are used in proving the uniqueness of the solution to the full moment problem. As for the existence of a solution, constrained extension results for linear operators were reviewed in [29], where the relationship between the moment problem and other topics on convexity have been discussed. Finally, among other results, article [30] solves concrete one-dimensional operator-valued moment problems, using the generalized Hahn–Banach result and the polynomial approximation result from [28], on the entire interval $\left[0,\infty \right).$ In the present review paper, results of [30] are improved and completed. Our conditions can be expressed in terms of limits of quadratic forms, since the involved approximating polynomials are nonnegative on $\left[0,\infty \right).$ The constraints on the solution are expressed by means of restrictions of linear operators to the positive cone of the domain space. Results on applied approximation and expansions have been recently published in references [31,32]. The rest of the paper is organized as follows. Section 2 summarizes the main methods applied in this work. In Section 3, the results are stated and proved. In Section 4, a brief discussion on the results is added.

2. Methods

Next, we summarize the methods used in this work.

• Using general convexity-type results to deduce or suggest the proofs of theorems on convex optimization. The same general results lead to the notion of integral representation (Theorems 2 and 3 below) and to that of the barycenter of a probability measure on a compact subset of a locally convex space. These further yield Jensen inequality in this framework (Theorem 8 below).
• Using general Hahn–Banach-type results for operators to obtain necessary and sufficient conditions for the existence of the linear solution $T$ satisfying the conditions (3) and (4), as written above. It is worth noticing that in the second inequality (4), $P:X_{+}⟶Y$ is a convex operator, which might not necessarily be sublinear.
• Using an earlier Hahn–Banach-type result on the extension of linear operators preserving positivity (recently recalled in Lemma 1.5 from [14]) to prove the uniqueness of the solution of the full moment problem on the nonnegative semi-axes. This is done via the polynomial approximation result on $\left[0,+\infty \right)$ of any element from ${\left(L_{\mu }^1\left(\left[0,+\infty \right)\right)\right)}_{+}$ (where $\mu$ is (M)-determinate [22]) by nonnegative polynomials on $\left[0,+\infty \right)$ (see Lemma 4.11 from [29]). For uniform approximation on compact subsets in $\left[0,+\infty \right)$ by nonnegative polynomials on $\left[0,+\infty \right)$, see Lemma 2 from [28]. Theorem 12 below uses this last-mentioned result and the expression of nonnegative polynomials on $\left[0,+\infty \right)$ in terms of sums of squares, recalled in [14].
• Using functional calculus for continuous functions (which preserves inequalities) [6], on the spectrum of a self-adjoint operator acting on a real or complex Hilbert space. To this aim, it is convenient to work in the order-complete Banach lattice of self-adjoint operators constructed in [7], pp. 303–305. It was applied in several previous papers, such as the recent article [29]. We recall that this space is also commutative algebra.
• Improving and completing the results of the article [30].

3. Results

3.1. An Application of the Krein–Milman Theorem to Completely Monotonic Functions

The aim of this first subsection is to recall Bernstein’s integral representation theorem of completely monotonic functions and related results. We start by recalling the statement of the Krein–Milman theorem. In what follows, all the locally convex spaces are Hausdorff. We are using results and ideas from [1,2,3,4,5,6,8].

Theorem 1 (Krein–Milman).

Any convex compact subset of a locally convex space is the closed convex hull of its set of extreme points.

For the proof of Theorem 1, see [1] or [3]. In the sequel, the terminology and the way of presentation follows the book [2] and the review paper [29]. Variants and consequences of a Krein–Milman-type theorem for finite dimensional compact and convex subsets can be found in [2,3,29].

Theorem 2 (Choquet).

Suppose that $K$ is a metrizable compact convex subset of the local convex space $X$, and that $x_0$ is an element of $K$. Then there is a probability measure $m$ on $K$, which represents $x_0$ and is supported by the extreme points of $K$.

This means for every continuous linear functional $L$ on $X,$ we have the following:

$$L\left(x_0\right)={\int }_{Ex\left(K\right)}Ldm,$$

where Ex$\left(K\right)$ is the set of extreme points of K. Next, we recall that a real valued function $f$ on $\left(0,+\infty \right)$ is called completely monotonic if it has derivatives of all natural orders and ${\left(-1\right)}^n{f}^{\left(n\right)}\ge 0$ on $\left(0,+\infty \right)$ for all natural numbers $n=\mathrm{0,1},2, \dots$ Basic examples of such functions are the following: $e^{-\alpha t}, t^{-\alpha }, \alpha \ge 0,$ and $t\in \left(0,+\infty \right)$. Hence, any completely monotonic function is nonincreasing and nonnegative on the entire interval $\left(0,+\infty \right)$ and so is each of its derivatives of even orders. The set of all such functions $f,$ with $f\left(0+\right)<+\infty$ is a convex cone, which will be denoted by CM. Obviously, since each function from CM is nonincreasing and nonnegative, the condition $f\left(0+\right)<+\infty$ is equivalent to the boundedness of $f$ on $\left(0,+\infty \right).$ The following theorem gives the integral representation of bounded completely monotonic functions. In what follows, $\left[0,+\infty \right]$ denotes the one–point compactification of the locally compact metric space $\left[0,+\infty \right).$ The following theorem gives the integral representation of bounded completely monotonic functions. In what follows, $\left[0,+\infty \right]$ denotes the one-point compactification of the locally compact metric space $\left[0,+\infty \right).$ We denote by $X$ the real vector space of all real valued infinitely differentiable functions on $\left(0,+\infty \right).$ One endows $X$ with the locally convex topology of uniform convergence of functions with all their derivatives on compact subsets of $\left(0,+\infty \right).$ This topology is generated by the sequence of pseudo-norms defined by the following:

$$p_{m,n}\left(f\right)=sup\left\{\left|f^k\left(t\right)\right|: m^{-1}\le t\le m, 0\le k\le n\right\},\left(m,n=\mathrm{1,2},3,\dots \right).$$

Then $X$ is metrizable and every bounded closed subset of $X$ is compact. Let $K$ be the convex subset of $CM$ formed by all $f\in CM$ with $f\left(0+\right)\le 1.$ If $f\in CM,$ then $f/f\left(0+\right)\in K,$ so it is sufficient to prove the theorem for $f\in K.$ The next idea is to apply the integral representation provided by Theorem 2 stated above, via the Lemmas 1 and 2 stated below.

To prove the next theorem, (Theorem 3) we recall the following lemmas.

Lemma 1.

The convex subset $K$ is compact in the local convex metrizable space $X$, as mentioned above.

Lemma 2.

Let us denote by $Ex\left(K\right)$ the set of all extreme functions of $K$. Then $f\in Ex\left(K\right)$ if and only if

$$f\left(t\right)=e^{-\alpha t}, t>0, 0\le \alpha \le +\infty .$$

(5)

Theorem 3 (Bernstein).

If $f$ is bounded and completely monotonic on the interval $\left(0,+\infty \right)$, then there exists a unique nonnegative Borel measure $\mu$ on $\left[0,+\infty \right]$ such that $\mu \left([0,+\infty ]\right)=f\left(0+\right)$, and for each $t>0$,

$$f\left(t\right)={\int }_0^{+\infty }{e}^{-\alpha t}d\mu \left(\alpha \right).$$

For the complete proof of Theorem 3, see [2]. See also, [8].

Proof (the sketch). 

The proof can be made using the representation provided by Theorem 2 stated above, via Lemmas 1 and 2 stated above.

Obviously, for $\alpha =0$ we have the corresponding constant function $f_0\left(t\right)=1, t>0,$ and for $\alpha =+\infty$, one obtains the constant function $f_{(+\infty )}\left(t\right)=0, t>0$.

Now we return to the idea of the proof of Theorem 3. We have already seen that it is sufficient to prove it for $f\in K.$ According to Theorem 2, for such a function $f,$ there exists a Borel probability measure $m$ on $Ex\left(K\right)$ such that $L\left(f\right)={\int }_{Ex\left(K\right)}Ldm$ for all linear continuous functionals $L$ on $X.$ For $t>0,$ we consider the linear functional $L_t\left(f\right)=f\left(t\right),$ so that

$$f\left(t\right)=L_t\left(f\right)={\int }_{Ex\left(K\right)}{L}_{t}dm, f\in K.$$

On the other hand, the map $T:\alpha \to e^{-\alpha \left(·\right)}$ from $\left[0,+\infty \right]$ into $K$ is continuous, so that its image $Ex\left(K\right)$ through it is compact. Since $L_t\left(T\alpha \right)=e^{-\alpha t},$ we have:

$$f\left(t\right)={\int }_{Ex\left(K\right)}{L}_{t}dm={\int }_{T^{-1}\left(Ex\left(K\right)\right)}\left(L_t\circ m\right)d\left(m\circ T\right)={\int }_0^{+\infty }{e}^{-\alpha t}d\mu \left(\alpha \right), t>0.$$

(6)

From this last expression of $f\left(t\right),$ we derive:

$$f\left(0+\right)={\int }_0^{+\infty }d\mu \left(\alpha \right)=\mu \left(\left[0,+\infty \right]\right).$$

(7)

For fixed $t\ge 0,$ the uniqueness of the measure $\mu$ follows via Stone-Weierstrass theorem on the density of the subalgebra generated by the continuous functions $\alpha \to e^{-\alpha t}$ on $\left[0,+\infty \right],$ in the space $C\left(\left[0,+\infty \right]\right).$ This ends the ideas of the proof. □

Remark 1.

The converse of Theorem 3 holds too. Namely, if $f$ is given by Equation (6), then differentiation under the integral sign with respect to the parameter $t$, leads to the following:

$${\left(-1\right)}^n{f}^{\left(n\right)}\left(t\right)={\left(-1\right)}^{2n}{\int }_0^{+\infty }{\alpha }^n{e}^{-\alpha t}d\mu \left(\alpha \right)={\int }_0^{+\infty }{\alpha }^n{e}^{-\alpha t}d\mu \left(\alpha \right)\ge 0, t>0, n=\mathrm{1,2},3,\dots$$

Hence, $f$ is completely monotonic, and (7) holds.

Lemma 3 (see [1]).

If K is a convex compact subset in the locally convex space, then every closed real hyperplane supporting K contains at least one extreme point of K.

Theorem 4 (Bauer) (see [2,3]).

Any convex continuous real-valued function defined on a compact convex subset K of a locally convex space, attains its maximum value (not necessarily uniquely) at an extreme point of K.

Theorem 4 provides a method for determining the maximum value and at which point(s) this value is attained of a continuous convex function on a compact subset $K$ whose extreme points are known. The simplest case is that of a subset $K$ whose subset $Ex\left(K\right)$ of all extreme points is finite. The next evaluation problem refers to finding a lower bound for such a function or for integrals of such a function with respect to a positive Borel measure. Regarding the latter problematic, an answer is provided by the integral form of Jensen inequality. Namely, we recall the following result, stated and proved in chapters dedicated to convexity in several basic books. The terminology and the way of presentation is that of the reference [5]. The next optimization type problem for a convex function is the following: what can we say about the minimum point of a real-valued convex continuous function $f$ on a compact convex subset $K$? Under usual conditions on $K,$ the minimum on $K$ is attained at a point $x_0$ located in the relative interior of $K$ if and only if $0\in 𝜕f\left(x_0\right)$. We recall that $𝜕f\left(x_0\right)$ is the set of all subgradients $T$ of $f$ at $x_0$. A linear functional $T$ on a locally convex space is called a subgradient of $f$ at $x_0$ if $f\left(x\right)-f\left(x_0\right)\ge T\left(x-x_0\right)$ for all points $x\in K.$ Hence, if $T=0,$ it results the desired conclusion $f\left(x\right)-f\left(x_0\right)\ge 0$ for all $x\in K.$ The next question is the following: what happens when the minimum is attained on the boundary $𝜕K?$ In this case, the existence of at least one continuous affine function $h$ dominated by $f$ on $K$ gives the answer. The method works for convex operators $P$ defined on arbitrary convex bounded (not necessarily compact) subsets $B\subset {\mathbb{R}}^n$, the codomain if $P$ being an order-complete vector lattice. This result has been quite recently reviewed in [29], Theorem 5.1. There the proof of this old result was also recalled, showing that $P$, (which stands for $f$ mentioned above), is bounded from below. The property that every convex function on a convex subset $B$ of a vector space is bounded below, characterizes the finite dimensionality property for $B.$ With the above notations, if $B\subset co\left\{v_1,\dots ,v_k\right\}$ for some $v_i$ and $i=1,\dots ,k$, then

$$P\left(x\right)=P\left({\sum }_{i=1}^k{\alpha }_i{v}_i\right)\ge h\left({\sum }_{i=1}^k{\alpha }_i{v}_i\right)={\sum }_{i=1}^k{\alpha }_{i}h\left(v_i\right)\ge inf\left\{h\left(v_i\right):1\le i\le k\right\}$$

for all ${\alpha }_i\in \left[0,+\infty \right),$ with ${\sum }_{i=1}^k{\alpha }_i=1.$ Hence, in this case, $P\left(x\right)\ge inf\left\{h\left(v_i\right):1\le i\le k\right\}$ for all $x\in B$.

Theorem 5 (Jensen’s Inequality).

Let $\mu$ be a probability measure on a $\sigma -$algebra $\mathfrak{M}$ in a set $\Omega .$ If $f$ is a real function in $L^1\left(\mu \right)$ such that $a<f\left(x\right)<b$ for all $x\in \Omega$, and if $\phi$ is real-valued and convex on $\left(a,b\right)$, then

$$\phi \left({\int }_{\Omega }fd\mu \right)\le {\int }_{\Omega }\left(\phi \circ f\right)d\mu .$$

See also Theorem 8 below.

Remark 2.

The cases $a=-\infty$ and $b=+\infty$ are not excluded. On the other hand, it may happen that $\phi \circ f$ is not an element of the space $L^1\left(\mu \right)$. According to the basic measure theory results and the proof from [5] of Theorem 5 stated above, the integral of $\phi \circ f$ exists in the extended sense and equals $+\infty$.

Our next aim is to recall a Jensen type inequality referring to the notion of the barycenter of a probability measure on a compact convex subset $K$ in a locally convex space $X.$ Starting with a probability measure supported by a finite set in the $\left\{x_i;i=\mathrm{1,2},\dots ,N\right\}\subset K$ and associating to each point $x_i$ the mass, $m_i\ge 0$ for all $i=1,\dots ,N, {\sum }_{i=1}^N{m}_i=1$.

We consider the measure $d\mu ≔{\sum }_{i=1}^N{m}_i{\delta }_{x_i},$ where ${\delta }_{x_i}\left(f\right)≔f\left(x_i\right), i=1,\dots ,N,$ with $f$ being a continuous real function on $K.$ Then for a linear continuous functional $L$ on $X,$ we can write the following:

$${\int }_{K}Ld\mu =\mu \left(L\right)≔{\sum }_{i=1}^N{m}_i{\delta }_{x_i}\left(L\right)={\sum }_{i=1}^N{m}_{i}L\left(x_i\right)=L\left({\sum }_{i=1}^N{m}_i{x}_i\right)=L\left(b_{{\mu }_{.}}\right).$$

(8)

The element $b_{\mu }≔{\sum }_{i=1}^N{m}_i{x}_i\in co\left\{x_1,\dots ,x_N\right\}$ is called the barycenter of the measure $\mu$ and verifies (8) for all $L\in X^{*}$, where $X^{*}$ is the linear topological dual of the locally convex space $X.$ In other words, $X^{*}$ consists of all linear continuous functionals $L:X⟶\mathbb{R}.$ Hence, any probability measure having as support a finite set $\left\{x_i;i=\mathrm{1,2},\dots ,N\right\}\subset K$ has barycenter $b_{\mu }\in co\left\{x_1,\dots ,x_N\right\}.$ The following much more general result holds (see [2,29]).

Definition 1.

Suppose that K is a nonempty compact subset of a locally convex space X and μ is a probability measure on K. We say that point $b_{\mu } in X is the barycenter of \mu$ if

$$L\left(b_{\mu }\right)={\int }_{K}Ld\mu$$

for every continuous linear functional $L$ on $X$.

Theorem 6.

If K is a compact convex subset in the locally convex space X and μ is a probability measure on K, there exists a unique barycenter $b_{\mu }\in K of \mu$.

Theorem 7.

Every point of a compact convex subset K in a locally convex space X is the barycenter of a Borel probability measure on K that is supported by the closure of the extreme points of K.

The proof of Theorem 7 is given in [3]. The next result represents the Jensen integral inequality for a barycenter and for probability measures (see [3]).

Theorem 8 (Jensen).

Suppose that dμ is a probability measure on the convex compact subset K of the locally convex space X and let $b_{\mu }$ be the barycenter of dμ. Then

$$f\left(b_{\mu }\right)\le {\int }_{K}f\left(x\right)d\mu \left(x\right)$$

for all continuous convex functions $f:K\to \mathbb{R}.$

3.2. On Simplexes and Related Results

The definition and basic properties of finite dimensional simplexes can be found in [3]. The notion of an infinite dimensional simplex is strongly related to the problem of the uniqueness of the representing measure from the Choquet integral representation theorem stated above as Theorem 2 (without uniqueness condition). For information on infinite dimensional simplexes see [2,3,8]. Results on finite simplicial unbounded sets have been reviewed in [29]. The detailed proofs for these results can be found in the corresponding original articles, cited in the references lists. In the end of this subsection, we recall a geometrical characterization for infinite dimensional simplexes. The statement of this result does not use the notion and results on integral representation or ordering.

Definition 2.

Let $K$ be a compact convex subset in the locally convex space $X$ A homothetic image of $K$ is any set of the form $\alpha K+x$, where $\alpha >0$ and $x\in X.$

Theorem 9.

A compact convex subset $K$ in a locally convex space $X$ is a simplex if and only if the intersection

$$\left(\alpha K+x\right)\cap \left(\beta K+y\right)$$

of any two homothetic images of $K$  is either empty, or a single point, or another homothetic image of  $K.$

Example 1.

A homothetic image of any convex triangle $K\subset {\mathbb{R}}^2$ is a triangle $T$ having the corresponding edges pairwise parallel to the edges of the triangle $K$. (Hence, the angles are pairwise equal). The intersection of any two tringles both being homothetic to $K$ is a triangle homothetic to $K.$ However, if K is (convex) square, the intersection appearing in Theorem 9 could be a convex rectangle with adjacent edges of inequal lengths. Such a rectangle is not homothetic to a square. Moreover, in a convex triangle T, any point of T can be uniquely represented by a convex combination of the vertices. This is no longer valid for a convex square or any other convex polygon. This statement confirms the geometric intuition that the intersection of two homothetic images of a triangle K will “look” just like $K.$ This case occurs unlike that of an arbitrary compact convex subset $K.$ For example, if $K$ is a closed disc, a homothetic image is a disc as well. However, the intersection of two discs does not look like a disc. In [12], a proof of Theorem 9 that avoids the compactness assumption was given.

3.3. Applications of Hahn–Banach-Type Theorems and Polynomial Approximation to the Moment Problem

In what follows, one applies a generalized version of the Hahn–Banach theorem and polynomial approximation on unbounded subsets, recalled in [28], as well as recent results from [29], to solve operator-valued moment problems. Let $A$ be a positive invertible self-adjoint operator acting on the real or complex Hilbert space $H$, and let $Y\left(A\right)$ be the commutative algebra over the real field of the self-adjoint operators constructed in [7], pp. 303–305. This algebra is a good example of subspace of the ordered Banach space $\mathcal{A}=\mathcal{A}\left(H\right)$ of all self-adjoint operators from $H$ into $H,$ endowed with the natural order relation

$$A\le B \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if}<Ah;h> \le <Bh;h> \mathrm{for} \mathrm{all} h\in H.$$

Endowed with the operational norm, $Y\left(A\right)$ is a Banach lattice that is also order-complete. The definition and a few results on this order-complete Banach lattice has been reviewed in [29], pp. 8–9. We denote by $int{\left(Y\left(A\right)\right)}_{+}$ the interior of the positive cone ${\left(Y\left(A\right)\right)}_{+}$. It is easy to observe that $int{\left(Y\left(A\right)\right)}_{+}$ consists of all the operators $U\in Y\left(A\right)$ for which there exists $\epsilon >0$ such that $U\ge \epsilon I.$ Here, $I:H⟶H$ is the identity operator.

Theorem 10.

Let $A$ be positive self-adjoint operator from $H$ into $H$. We denote

$$p_j\left(t\right)≔t^j, t\in \left[0,+\infty \right), j\in \mathbb{N}.$$

Let ${\left(E_j\right)}_{j\in \mathbb{N}}$ be is a sequence in $Y\left(A\right),$  and $P:{\left(C\left(\sigma \left(A\right)\right)\right)}_{+}⟶Y\left(A\right)$ be defined by

$$\tilde{P}\left(g\right)≔{\left(I+g\left(A\right)\right)}^{-1}$$

The following statements are equivalent:

(i) 

There exists a unique positive linear operator $T:C\left(\sigma \left(A\right)\right)⟶Y\left(A\right)$ satisfying the moment conditions

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

with

$$T\left(g\right)\le {\left(I+g\left(A\right)\right)}^{-1} for all g\in {\left(C\left(\sigma \left(A\right)\right)\right)}_{+}, ‖T‖\le 1/2.$$

(ii) 

For any finite subset $J_0\subset \mathbb{N},$ any set of scalars ${\left\{{\alpha }_j\right\}}_{j\in J_0}$, the following implication holds. If

$$\sum _{j\in J_0}{\alpha }_j{t}^j\le g\left(t\right)$$

for all $t\in \sigma \left(A\right)$, then

$$\sum _{j\in J_0}{\alpha }_j{E}_j\le {\left(I+g\left(A\right))\right)}^{-1}$$

Proof. 

We can apply Theorem 1 from [19] or Theorem 2.30 from [29], pp. 373–374. Namely, in the latter theorem, $E$ stands for $C\left(\sigma \left(A\right)\right)$, $M$ is the subspace of $E$ consisting of all the polynomial functions with real coefficients, $F$ stands for $Y\left(A\right)$, $P:$ ${\left(C\left(\sigma \left(A\right)\right)\right)}_{+}⟶Y\left(A\right)$, and $P\left(g\right)≔{\left(I+g\left(A\right)\right)}^{-1}$ for all $g\in {\left(C\left(\sigma \left(A\right)\right)\right)}_{+}.$ To apply the invoked theorem, we must prove that $P$ is convex on ${\left(C\left(\sigma \left(A\right)\right)\right)}_{+}.$ Since $g\in {\left(C\left(\sigma \left(A\right)\right)\right)}_{+}$ and

$$\Psi :C\left(\sigma \left(A\right)\right)\to Y\left(A\right), \Psi \left(g\right)≔g\left(A\right)$$

is a linear positive operator from $C\left(\sigma \left(A\right)\right)$ into $Y\left(A\right),$ we have

$$g\left(A\right)=\Psi \left(g\right)\in {\left(Y\left(A\right)\right)}_{+}$$

On the other hand, the image $g\left(\sigma \left(A\right)\right)$ of the spectrum $\sigma \left(A\right)$ through the real valued nonnegative function $g\in {\left(C\left(\sigma \left(A\right)\right)\right)}_{+}$ is contained in $\left[0,+\infty \right).$ These further yield

$$\sigma \left(I+g\left(A\right)\right)=1+g\left(\sigma \left(A\right)\right)\subset \left[1,+\infty \right), I+g\left(A\right)\ge I$$

Thus, zero is not an element of $\sigma \left(I+g\left(A\right)\right)$. This means that $I+g\left(A\right)=I+\Psi \left(g\right)$ is invertible. In other words, there exists ${\left(I+g\left(A\right)\right)}^{-1}.$ We can also write

$$\sigma \left({\left(I+g\left(A\right)\right)}^{-1}\right)={\left(1+g\left(\sigma \left(A\right)\right)\right)}^{-1}\subset \left(\mathrm{0,1}\right], {\left(I+g\left(A\right)\right)}^{-1}\le I$$

$$‖{\left(I+g\left(A\right)\right)}^{-1}‖=sup\left(\sigma \left({\left(I+g\left(A\right)\right)}^{-1}\right)\right)\le 1, \forall g\in {\left(C\left(\sigma \left(A\right)\right)\right)}_{+}.$$

All these remarks lead to a first conclusion: the operator $P$ defined at the beginning of the proof is well defined on ${\left(C\left(\sigma \left(A\right)\right)\right)}_{+}$, and $P\left(g\right)\in int{\left(Y\left(A\right)\right)}_{+}=\left\{U\in {\left(Y\left(A\right)\right)}_{+}: \exists U^{-1}\in {\left(Y\left(A\right)\right)}_{+}\right\}$ for all $g\in {\left(C\left(\sigma \left(A\right)\right)\right)}_{+}$. Thus, to prove the convexity of $P,$ it is sufficient to show that the mapping $\Phi :int{\left(Y\left(A\right)\right)}_{+}⟶int{\left(Y\left(A\right)\right)}_{+}$ and

$$\Phi \left(B\right)≔B^{-1}, B\in int{\left(Y\left(A\right)\right)}_{+}$$

are convex on $int{\left(Y\left(A\right)\right)}_{+}$. Assuming this is conducted, $P\left(g\right)=\Phi \left(I+g\left(A\right)\right)={\left(I+g\left(A\right)\right)}^{-1}$ is convex, as a composition of the convex operator $\Phi$ with the affine operator

$$W:{\left(C\left(\sigma \left(A\right)\right)\right)}_{+}⟶{\left(Y\left(A\right)\right)}_{+}, W\left(g\right)=I+g\left(A\right)=I+\Psi \left(g\right).$$

To prove that $\Phi$ is convex, we must show that for any $U,V\in int{\left(Y\left(A\right)\right)}_{+}$ we have

$${\left(\left(1-\alpha \right)U+\alpha V\right)}^{-1}\le \left(1-\alpha \right)U^{-1}+\alpha V^{-1},$$

(9)

for all $\alpha \in \left[\mathrm{0,1}\right].$ Since $U,V$ are commuting and are positive self-adjoint operators, multiplying in (12) by $V={\left(V^{-1}\right)}^{-1},$ the inequality (12) is equivalent to

$${\left(\left(1-\alpha \right)U{V}^{-1}+\alpha I\right)}^{-1}\le \left(1-\alpha \right)({U{V}^{-1})}^{-1}+\alpha I, \alpha \in \left[\mathrm{0,1}\right].$$

(10)

Now (13) follows via functional calculus for the self-adjoint positive invertible operator $U{V}^{-1},$ from the convexity of the elementary function $\phi \left(t\right)≔t^{-1}, t\in \left(0,+\infty \right),$ as follows:

$${\left(\left(1-\alpha \right)t+\alpha \right)}^{-1}\le \left(1-\alpha \right)t^{-1}+\alpha , \alpha \in \left[\mathrm{0,1}\right], t\in \sigma \left(U{V}^{-1}\right)\subset \left(0,+\infty \right).$$

For $\phi$ contained in the closed unit ball of the space $C\left(\sigma \left(A\right)\right),$ we have $\left|\phi \right|\le \mathbf{1}$. The positivity of the linear operator $T,$ the preceding remark, and inequality (11) imply

$$\left|T\left(\phi \right)\right|\le T\left(\left|\phi \right|\right)\le T\left(\mathbf{1}\right)\le P(\mathbf{1})\le {\left(I+I)\right)}^{-1}={\displaystyle \frac{1}{2}}I.$$

Using the property of the norm on the Banach lattice $Y\left(A\right),$ this implies $‖T\left(\phi \right)‖\le {\displaystyle \frac{1}{2}}‖I‖={\displaystyle \frac{1}{2}}$ for all $\phi$ from the Banach lattice $C\left(\sigma \left(A\right)\right)$, with $‖\phi ‖\le 1$. □

Theorem 11.

In the framework and using the notations from Theorem 10 proved above, let $n\ge 2$ be an integer. The following statements are equivalent:

(i) 

There exists a unique positive linear operator $T:C\left(\sigma \left(A\right)\right)⟶Y\left(A\right)$ satisfying the moment conditions:

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

with

$$T\left(g\right)\le {\left(I+g\left(A\right)\right)}^{-n} for all g\in {\left(C\left(\sigma \left(A\right)\right)\right)}_{+}, ‖T‖\le 2^{-n}.$$

(ii) 

For any finite subset $J_0\subset \mathbb{N},$ any set of scalars ${\left\{{\alpha }_j\right\}}_{j\in J_0}$, the following implication holds. If

$$\sum _{j\in J_0}{\alpha }_j{t}^j\le g\left(t\right)$$

for all $t\in \sigma \left(A\right),$ then

$$\sum _{j\in J_0}{\alpha }_j{E}_j\le {\left(I+g\left(A\right)\right)}^{-n}.$$

Proof. 

The proof follows the same results and ideas as those from the proof of Theorem 10, and also uses the convexity of the operator $p_n:{\left(Y\left(A\right)\right)}_{+}⟶{\left(Y\left(A\right)\right)}_{+}, p_n\left(V\right)≔V^n.$ We repeat the proof of Theorem 10 provided above, where $P\left(g\right)≔{\left(I+g\left(A\right)\right)}^{-1}$ is replaced by $P_n\left(g\right)≔{\left(I+g\left(A\right)\right)}^{-n}.$ To finish the proof, it must be shown only the convexity of $P_n$ on ${\left(C\left(\sigma \left(A\right)\right)\right)}_{+}.$ This property is almost obvious, since we can write

$$P_n=p_n\circ P.$$

$P_n$ is convex and is obviously monotone increasing on ${\left(Y\left(A\right)\right)}_{+}.$ On the other hand, $P$ appearing in Theorem 10 proved above is convex. Hence $P_n$ is convex. □

Corollary 1.

If the sequence ${\left(E_j\right)}_{j\in \mathbb{N}}$ of self-adjoint operators verifies condition (ii) of Theorem 11, then

$$0\le E_j\le {\left(I+A^j\right)}^{-n}, j\in \mathbb{N}.$$

Proof. 

In both Theorems 10 and 11, the self-adjoint operator $A$ is assumed to be positive. Thus, its spectrum $\sigma \left(A\right)$ is contained in the interval $\left[0,+\infty \right).$ Therefore, the basic polynomials $p_j, p_j\left(t\right)=t^j$ have nonnegative values at all points $t\in \sigma \left(A\right)$. Since ${\left(E_j\right)}_{j\in \mathbb{N}}$ verifies (ii) of Theorem 11, the equivalent condition (i) is also verified. Consequently, due to the positivity of the linear solution $T$ appearing in (i), we infer that:

$$E_j=T\left(p_j\right)\ge 0, E_j=T\left(p_j\right)\le {\left(I+p_j\left(A\right)\right)}^{-n}={\left(I+A^j\right)}^{-n}, j\in \mathbb{N}.$$

□

Our next result, proved below, refers to the scalar valued moment problem on the unbounded interval $\left[0,+\infty \right).$ Unlike the previous theorems of the present review paper, now a polynomial approximation result in $L_{\mu }^1\left([0,+\infty )\right)$ is applied. Here $\mu$ is a positive regular Borel (M) determinate measure on $\left[0,+\infty \right),$ namely $d\mu =e^{-t}dt.$ For this measure, the corresponding classical moments ${\int }_0^{\infty }{t}^n{e}^{-t}dt$, $n\in \mathbb{N}$ are well-known and easily be computed, integrating by parts successively (or using an elementary property of the Euler Gamma function).

Theorem 12.

Let ${\left(m_n\right)}_{n\ge 0}$ be a sequence of real numbers, $d\mu =e^{-t}dt, t\in \left[0,+\infty \right)$. The following statements are equivalent:

(a) 

There exists a unique $f\in L_{\mu }^{\infty }\left([0,+\infty )\right)$, $0\le f\le 1$ in $L_{\mu }^{\infty }\left([0,+\infty )\right)$, such that

$${\int }_0^{+\infty }{t}^{n}f\left(t\right)e^{-t}dt=m_n, n\in \mathbb{N}.$$

(11)

(b) 

For any finite subset $J\subset \mathbb{N}$, any $\left\{{\alpha }_j;j\in J\right\}\subset \mathbb{R}$, and $k\in \left\{\mathrm{0,1}\right\}$ the following inequalities hold:

$$0\le \sum _{i,j\in J}{\alpha }_i{\alpha }_j{m}_{i+j+k}\le \sum _{i,j\in J}{\alpha }_i{\alpha }_j\left(i+j+k\right)!.$$

(12)

Proof. 

We apply the Hahn–Banach-type result named in the beginning of the proof of Theorem 10 above, to

$$E≔L_{\mu }^1\left(\left[0,+\infty \right)\right),M≔\mathbb{R}\left[t\right], Y≔\mathbb{R},$$

$$T_0:M⟶\mathbb{R},T_0\left(\sum _{j\in F}{\alpha }_j{p}_j\right)≔\sum _{j\in F}{\alpha }_j{m}_j, P:X_{+}.⟶\mathbb{R},$$

$$P\left(g\right)≔{\int }_0^{+\infty }g\left(t\right)e^{-t}dt, g\in E_{+}.$$

We observe that $P$ is the restriction to $E_{+}$ of the linear positive functional

$$T_2:E⟶\mathbb{R},T_2\left(h\right)≔{\int }_0^{+\infty }h\left(t\right)e^{-t}dt={\int }_0^{+\infty }hd\mu , h\in E.$$

(13)

To prove the main implication (b) implies (a), we observe that by the first inequality (15), one has

$$T_0\left(\sum _{j\in J_0}{\alpha }_j{p}_j\right)\ge 0,$$

for all nonnegative polynomials $p=\sum _{j\in J_0}{\alpha }_j{p}_j\ge 0$ on $\left[0,+\infty \right).$ Here $J_0$ is an arbitrary finite subset of the set $\mathbb{N}≔\left\{\mathrm{0,1},2,\dots \right\}$ of natural numbers. This assertion holds true due to the expression of any such polynomial as a finite sum of polynomials of the form $\sum _{i,j\in J}{\alpha }_i{\alpha }_j{p}_{i+j+k}, k\in \left\{\mathrm{0,1}\right\}$ (see [14]). By the Haviland theorem, there exists a positive Radon measure $v$ supported on $\left[0,+\infty \right),$ such that

$$T_0\left(p\right)={\int }_0^{+\infty }pd\nu , p\in \mathbb{R}\left[t\right].$$

(14)

On the other hand, according to the Stieltjes case for sufficient conditions for determinacy [22], $d\mu =e^{-t}dt$ is a moment determinate measure on $\left[0,+\infty \right),$ with $\mu \left(\left[0,+\infty \right)\right)={\int }_0^{+\infty }{e}^{-t}dt=1<+\infty .$ Now for any $\phi \in {\left(L_{\mu }^1\left[0,+\infty \right)\right)}_{+},$ this ensures the existence of a sequence ${\left(q_m\right)}_{m\ge 0}$ of nonnegative polynomials $q_m⟶\phi$ in $L_{\mu }^1\left[0,+\infty \right),$ as $m⟶+\infty$. Indeed, according to [4,5], any $\phi \in {\left(L_{\mu }^1\left[0,+\infty \right)\right)}_{+}$ can be approximated by nonnegative compactly supported functions $\psi \in {\left(C_c\left[0,+\infty \right)\right)}_{+}.$ On the other hand, any such function $\psi \in {\left(C_c\left[0,+\infty \right)\right)}_{+}$ can be approximated by nonnegative polynomials $q_m, q_m\left(t\right)\ge 0, \forall t\in \left[0,+\infty \right),$ in the norm ${‖·‖}_1$ on $\left(L_{\mu }^1\left[0,+\infty \right)\right),$ since $\mu$ is moment determinate (see [29], Lemma 4.11). Hence, the positive polynomials on $\left[0,+\infty \right)$ are dense in ${\left(L_{\mu }^1\left[0,+\infty \right)\right)}_{+}.$ Summing inequalities of the same sense and looking to the hypothesis (b), we see those inequalities (15) lead to

$$0\le T_0\left(q_m\right)\le T_2\left(q_m\right)={\int }_0^{+\infty }{q}_{m}d\mu , m\in \mathbb{N}.$$

(15)

On $\mathbb{R}\left[t\right],$ the positive linear functional $T_0$ satisfies the conditions

$$0\le T_0\left(q\right)={\int }_0^{+\infty }qd\nu \le T_2\left(q\right)={\int }_0^{+\infty }qd\mu , q\left(t\right)\ge 0, t\in \left[0,+\infty \right), q\in \mathbb{R}\left[t\right].$$

So, if $\phi \in L_{\mu }^1\left[0,+\infty \right),\phi \left(t\right)\ge 0,t\in \left[0,+\infty \right),q_m\in \mathbb{R}\left[t\right],q_m\left(t\right)\ge 0,t\in \left[0,+\infty \right)$, and $q_m⟶\phi$ in $L_{\mu }^1\left[0,+\infty \right),$ then (19) holds for all $q_m,m\in \mathbb{N}$. Passing to the limit as $m⟶\infty ,$ it results

$$T\left(\phi \right)≔{\int }_0^{+\infty }\phi d\nu \le T_2\left(\phi \right)={\int }_0^{+\infty }\phi d\mu \mathit{for} \mathit{all} \phi \in {\left(L_{\mu }^1\left[0,+\infty \right)\right)}_{+}.$$

Thus, $T$ is a positive linear extension of $T_0$ to the entire space $E≔L_{d\mu }^1\left(\left[0,+\infty \right)\right).$ Writing this for $\phi ={\chi }_B$, with $B$ as an arbitrary Borel subset in $\left[0,+\infty \right)$ verifying $0<\mu \left(B\right),$ we find

$$\nu \left(B\right)={\int }_0^{+\infty }{\chi }_{B}d\nu \le {\int }_0^{+\infty }{\chi }_{B}d\mu =\mu \left(B\right).$$

It follows that there exists a Borel measurable function $f,0\le f\le \mathbf{1}$, such that $d\nu =fd\mu$ (see Theorems 6.10 and 1.40 of [5]). We claim that $f$ is the desired solution appearing at statement (a). It has already been mentioned that $f\in L_{\mu }^{\infty }\left(\left[0,+\infty \right)\right)$, $0\le f\le \mathbf{1}.$ To finish the sufficiency of the conditions (b), we verify the moment conditions (14) for the measure $d\nu =fd\mu =f\left(t\right)e^{-t}dt,$ as follows:

$${\int }_0^{+\infty }{t}^{n}f\left(t\right)d\mu ={\int }_0^{+\infty }{p}_n\left(t\right)d\nu =T_0\left(p_n\right)=m_n, n\in \mathbb{N}.$$

The uniqueness of the solution $f$ is a consequence of the density of polynomials in $L_{\mu }^1\left[0,+\infty \right)$. Due to the positivity of measures $d\mu ,d\nu ,$ and the nonnegativity of polynomials $t^k{\left(\sum _{j\in J_0}{\alpha }_j{t}^j\right)}^2,k\in \left\{\mathrm{0,1}\right\},$ and $t\in \left[0,+\infty \right)$, the converse implication (a) implies (b) is obvious. □

The moment problem solved by Theorem 12 and mainly by its proof, leads to the idea of the solution for a more general problem. In the next corollary, $\rho {\in L}_{dt}^1\left[0,+\infty \right),$ is continuous and bounded on $\left[0,+\infty \right),$ with $\rho \left(t\right)>0$ for all $t\in \left(0,+\infty \right)$.

Theorem 13.

Let $d\mu =\rho \left(t\right)dt$ be a moment determinate positive regular measure on $\left[0,+\infty \right),$ with finite moments of all natural orders. Let $f_i\in L_{\mu }^{\infty }\left[0,+\infty \right), i=\mathrm{1,2},$ be such that $0\le f_1\left(t\right)\le f_2\left(t\right)$ for almost all $t\in \left[0,+\infty \right)$. Let ${\left(y_j\right)}_{j\in \mathbb{N}}$ be a sequence of real numbers. We define $d\mu ≔\rho \left(t\right)dt$ and $U_i\left(\phi \right)≔{\int }_0^{\infty }(\phi ·f_i·\rho )\left(t\right)dt, i=\mathrm{1,2}, \phi \in {\in L}_{d\mu }^1\left[0,+\infty \right).$ The following statements are equivalent.

(a) 

There exists a unique function $f\in L_{\mu }^{\infty }\left[0,+\infty \right)$, $f_1\left(t\right)\le f\left(t\right)\le f_2\left(t\right)$ almost everywhere in $\left[0,+\infty \right),$ such that

$${\int }_0^{\infty }{t}^{j}d\nu ={\int }_0^{\infty }{t}^{j}f\left(t\right)d\mu =y_j, j\in \mathbb{N}.$$

(b) 

For any finite subset $J\subset \mathbb{N}$, any $\left\{{\alpha }_j;j\in J\right\}\subset \mathbb{R}$, and $k\in \left\{\mathrm{0,1}\right\},$ the following inequalities hold:

$$\sum _{i,j\in J}{\alpha }_i{\alpha }_j{\int }_0^{\infty }{t}^{i+j}{f}_1\left(t\right)d\mu \le \sum _{i,j\in J}{\alpha }_i{\alpha }_j{y}_{i+j+k}\le \sum _{i,j\in J}{\alpha }_i{\alpha }_j{\int }_0^{\infty }{t}^{i+j}{f}_2\left(t\right)d\mu .$$

Proof. 

As in the proof of Theorem 12, we define $U_0:M≔\mathbb{R}\left[t\right]⟶\mathbb{R},U_0\left(\sum _{j\in F}{\alpha }_j{p}_j\right)≔\sum _{j\in F}{\alpha }_j{y}_j,$ $J\subset \mathbb{N}$ is a finite subset, and $\left\{{\alpha }_j;j\in J\right\}\subset \mathbb{R}$ is an arbitrary (finite) subset. Conditions (b) of the present corollary say that

$${\int }_0^{\infty }{f}_1\left(t\right)t^k{r}^2\left(t\right)d\mu \le U_0\left(p_k{r}^2\right)\le {\int }_0^{\infty }{f}_2\left(t\right)t^k{r}^2\left(t\right)d\mu , r\in \mathbb{R}\left[t\right], k\in \left\{\mathrm{0,1}\right\}$$

Since any polynomial function $q,$ which is nonnegative at each point of the interval $[0,+\infty )$ is a finite sum of special polynomials $p_k{r}^2,r\in \mathbb{R}\left[t\right],k\in \left\{\mathrm{0,1}\right\}$ (see [14]), by summing the term in inequalities (20) we find that

$$0\le {\int }_0^{\infty }{f}_1\left(t\right)q\left(t\right)d\mu \le U_0\left(q\right)\le {\int }_0^{\infty }{f}_2\left(t\right)q\left(t\right)d\mu .$$

On the other hand, the convex cone of all such polynomials is dense in ${\left(L_{\mu }^1\left(\left[0,+\infty \right)\right)\right)}_{+},$ since $\mu$ is moment determinate (see [22,29]). If $\phi \in {\left(L_{\mu }^1\left(\left[0,+\infty \right)\right)\right)}_{+},$ let ${\left(q_m\right)}_m$ be a sequence of nonnegative polynomials on $\left[0,+\infty \right),$ such that $q_m⟶\phi$ in $L_{\mu }^1\left(\left[0,+\infty \right)\right).$ By what is already proved, we have

$${\int }_0^{\infty }{f}_1\left(t\right)q_m\left(t\right)d\mu \le U\left(\phi \right)\le {\int }_0^{\infty }{f}_2\left(t\right)q_m\left(t\right)d\mu , m\in \mathbb{N}.$$

Here, $U$ is the extension of $U_0$ obtained via Haviland theorem. Hence $U\left(\phi \right)={\int }_0^{\infty }f\left(t\right)\phi \left(t\right)\rho \left(t\right)dt$ for some Radon measurable nonnegative function $f.$ Passing to the limit as $m⟶\infty ,$ for $\phi ={\chi }_R$ (the characteristic function of an arbitrary Radon measurable subset $R$ contained $\left[0,+\infty \right)),$ we find

$${\int }_R\left(\phi \left(t\right)-f_1\left(t\right)\right)d\mu \ge 0, {\int }_R\left(f_2\left(t\right)-\phi \left(t\right)\right)d\mu \ge 0.$$

Writing these inequalities for all Bore subsets $B$ with $0<\mu \left(B\right)<\infty ,$ we conclude that $f_1\le \phi \le f_2$ almost everywhere in $\left[0,+\infty \right).$ Therefore, $f_1\le \phi \le f_2$ in $L_{\mu }^{\infty }\left([0,+\infty )\right)$ (in particular, $\phi \in L_{\mu }^{\infty }\left(\left[0,+\infty \right)\right).$ This proves the main implication (b) implies (a). The converse is obvious. □

Theorem 14.

Under the hypothesis of Theorem 13, assume that the subset $S\subset \left[0,+\infty \right)$ is compact, and $X≔C\left(S\right).$ Then the statements (a) and (b) from Theorem 13 are equivalent, and the solution $T$ has the additional property:

$$‖T_1\left(\mathbf{1}\right)‖\le ‖T‖=‖T\left(\mathbf{1}\right)‖\le ‖T_2\left(\mathbf{1}\right)‖$$

Proof. 

The implication (b) implies (a) is a consequence of Theorem 13, since any positive linear operator on Banach lattices is continuous. The converse implication is simple. Namely, the positivity of $T_1$ implies the positivity of $T$ and $T_2$ on $X_{+}$. These further yield

$$\left|T\left(g\right)\right|\le T\left(\left|g\right|\right)\le T\left({‖g‖}_{C\left(S\right)}·\mathbf{1}\right)={‖g‖}_{C\left(S\right)}·T\left(\mathbf{1}\right).$$

The monotone increasing property of the norm of $Y$ on the positive cone $Y_{+}$ leads to

$$‖T\left(g\right)‖=‖ \left|T\left(g\right)\right| ‖\le {‖g‖}_{C\left(S\right)}·‖T\left(\mathbf{1}\right)‖$$

This leads to: $‖T‖\le ‖T\left(\mathbf{1}\right)‖.$ Since the converse inequality is obvious, we conclude that $‖T‖=‖T\left(\mathbf{1}\right)‖$ for any positive linear operator $T:C\left(S\right)⟶Y.$ By the same reason, $‖T_i‖=‖T_i\left(\mathbf{1}\right)‖$, $i=\mathrm{1,2}$. Now a partial conclusion follows from inequalities (21) and equality $‖T‖=‖T\left(\mathbf{1}\right)‖$, which works for all positive linear operators $T.$ On the other hand, to prove the implication (a) implies (b), assuming that (a) holds true and

$$\left(q_m-r_m-\sum _{j\in J_0}{\lambda }_j{p}_j\right)⟶0, m⟶\infty$$

then

$$\sum _{j\in J_0}{\lambda }_j{y}_j=T\left(\sum _{j\in J_0}{\lambda }_j{p}_j\right)=\underset{m}{\mathit{lim}}T\left(q_m\right)-\underset{m}{\mathit{lim}}T\left(r_m\right)=$$

$$T\left({\phi }_2\right)-T\left({\phi }_1\right)\le T_2\left({\phi }_2\right)-T_1\left({\phi }_1\right)=\underset{m}{lim}(T_2\left(q_m\right)-T_1\left(r_m\right))$$

□

Next, we apply Theorems 13 and 14 to concrete spaces operators. Let $A$ be a self-adjoint operator acting on a Hilbert space $H,$ with the spectrum $\sigma \left(A\right)\subseteq \left[\mathrm{0,2}\right]$. We denote by $Y\left(A\right)$ the order-complete Banach lattice of self-adjoint operators constructed in [7], pp. 303–305. By $\Psi ={\Psi }_A:C\left(\sigma \left(A\right)\right)⟶Y\left(A\right)$ we denote the linear mapping $\Psi \left(\phi \right)≔\phi \left(A\right), \phi \in C\left(\sigma \left(A\right)\right),$ defined by the functional calculus [6,7] for continuous real-valued functions on the spectrum $\sigma \left(A\right)$. We denote $T_1\left(\phi \right)≔A·\mathit{log}\left(A\right)·\phi \left(A\right){, T}_2\left(\phi \right)≔A·\left(A-I\right)·\phi \left(A\right), \phi \in C\left(\sigma \left(A\right)\right)$.

Theorem 15.

Let ${\left(E_j\right)}_{j\in \mathbb{N}} be given a sequence of operators from Y\left(A\right).$ Let us consider the following two statements:

(a) 

There exists a unique bounded linear operator $T\in B(C\left(\sigma \left(A\right)\right),Y(A\left)\right)$ with

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

$$T_1\left(\phi \right)\le T\left(\phi \right)\le T_2\left(\phi \right), \phi \in {C\left(\sigma \left(A\right)\right)}_{+}$$

(b) 

If $J_0\subset \mathbb{N}$ is finite, $p_m$ and $q_m$ are nonnegative polynomials on $\left[0,\infty \right)$ and

$\left((t·\left(t-1\right)·p_m\left(t\right)-t·log\left(t\right){·q}_m\left(t\right)-\sum _{j\in J_0}{\lambda }_j{t}^j\right)⟶0, m⟶\infty$ uniformly on $\sigma \left(A\right)$, then

$$\sum _{j\in J_0}{\lambda }_j{E}_j\le \underset{m⟶\infty }{lim}(A·\left(A-I\right)·p_m\left(A\right)-A·log\left(A\right)·q_m\left(A\right)),$$

Under these conditions, (b) implies (a) holds.

Proof. 

We apply Theorem 13 for $S≔\sigma \left(A\right),X≔C\left(\sigma \left(A\right)\right),Y≔Y\left(A\right),y_j=E_j,j\in \mathbb{N},T_2\left(\phi \right)≔A·\left(A-I\right)·\phi \left(A\right),T_1\left(\phi \right)≔A·log\left(A\right)·\phi \left(A\right),\phi \in C\left(\sigma \left(A\right)\right).$ □

Remark 3.

Let us observe that both functions $f_1\left(t\right)=t·log\left(t\right), f_1\left(0\right)=f_1\left(0+\right)≔0$ and $f_2\left(t\right)=t\left(t-1\right)$ satisfy the conditions $\left(t\right)<0$ for all $t\in \left(\mathrm{0,1}\right), f_i\left(t\right)>0$ and for all $t\in \left(\mathrm{1,2}\right), f_i\left(0\right)=f_i\left(1\right)=0, f_i$ is continuous on $\left[\mathrm{0,2}\right],$ for $i=\mathrm{1,2}.$ Moreover, we can write

$$f_1\left(t\right)=t·log\left(t\right)=t·log\left(1+\left(t-1\right)\right)\le t\left(t-1\right)=f_2\left(t\right) for all t>0.$$

In the next result, we consider the operators $T_i:C\left(\sigma \left(A\right)\right)⟶Y\left(A\right), T_i\left(\phi \right)≔f_i\left(A\right)·\phi \left(A\right)$ for a self-adjoint operator $A$ whose spectrum is contained in $\left[\mathrm{1,2}\right], i=\mathrm{1,2}$. In this case, the two functions $f_1$ and $f_2$ are nonnegative on the spectrum $\sigma \left(A\right).$

Theorem 16.

Let $H$ be a Hilbert space, and $A:H⟶H$ a self-adjoint operator such that $\sigma \left(A\right)\subseteq \left[\mathrm{1,2}\right]$. Let $f_i, T_i, i=\mathrm{1,2}$ be as in Theorem 13. The following statements are equivalent.

(a) 

There exists a unique bounded linear operator $T\in B(X,Y(A\left)\right)$ with

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

$$T_1\left(\phi \right)\le T\left(\phi \right)\le T_2\left(\phi \right), \phi \in {C\left(\sigma \left(A\right)\right)}_{+}$$

$$‖A·log\left(A\right)‖\le ‖T‖\le ‖A·\left(A-I\right)‖$$

(b) 

If $J_0\subset \mathbb{N}$ is finite, $q_m$ and $r_m$ are nonnegative polynomials on $\left[0,\infty \right)$ and $Y\left(A\right)$ is as mentioned above, and if $\left(t·\left(t-1\right)·q_m\left(t\right)-t·log\left(t\right){·r}_m\left(t\right)-\sum _{j\in J_0}{\lambda }_j{t}^j\right)⟶0, m⟶\infty$ uniformly on $\sigma \left(A\right),$ then

$$\sum _{j\in J_0}{\lambda }_j{E}_j\le \underset{m⟶\infty }{lim}\left(A·\left(A-I\right)·q_m\left(A\right)-A·log\left(A\right)·r_m\left(A\right)\right)$$

Proof. 

We apply Theorem 14 for the involved positive linear operators $T_1, T_2$ defined in the proof of Theorem 15.

Using the preceding results, the following two theorems follow as well. □

Theorem 17.

Let $H, A, Y\left(A\right), {\left(E_j\right)}_{j\in \mathbb{N}}$ be as in Theorem 13, $f_1,f_2\in C\left(\sigma \left(A\right)\right), { f}_1\left(t\right)\le f_2\left(t\right)$ for all $t\in \sigma \left(A\right)$. Assume that $\sigma \left(A\right)\subseteq \left[0,\infty \right).$ Let $T_1, T_2$ be two bounded linear operators defined on $C\left(\sigma \left(A\right)\right)$, defined by $T_i\left(\phi \right):{=f}_i\left(A\right)·\phi \left(A\right), i=\mathrm{1,2},$ having $Y\left(A\right)$ as codomain.

Let us consider the following statements.

(a) 

There exists a unique bounded linear operator $T\in B(C\left(\sigma \left(A\right)\right),Y(A\left)\right)$ with

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

$$T_1\left(\phi \right)\le T\left(\phi \right)\le T_2\left(\phi \right), \phi \in {\left(C\left(\sigma \left(A\right)\right)\right)}_{+}.$$

(b) 

If $J_0\subset \mathbb{N}$ is finite, $q_m$ and $r_m$ are nonnegative polynomials on $\left[0,\infty \right)$ and $\left(\left(f_2\left(t\right)q_m\left(t\right)-f_1\left(t\right)r_m\left(t\right)-\sum _{j\in J_0}{\lambda }_j{t}^j\right)\right)⟶0, m⟶\infty$ uniformly on $\sigma \left(A\right),$ then

$$\sum _{j\in J_0}{\lambda }_j{E}_j\le \underset{m⟶\infty }{lim}\left(f_2\left(A\right)q_m\left(A\right)-f_1\left(A\right)r_m\left(A\right)\right)$$

Under these conditions, (b) implies (a) holds.

Theorem 18.

Under the hypothesis of Theorem 17, additionally assume that $0\le { f}_1\left(t\right)\le f_2\left(t\right) for all t\in \sigma \left(A\right).$ Then the conditions (a) and (b) of Theorem 17 are equivalent, $0\le T_1\left(\phi \right)\le T\left(\phi \right)\le T_2\left(\phi \right)$ for all $\phi \in {\left(C\left(\sigma \left(A\right)\right)\right)}_{+},$ and

$$‖f_1\left(A\right)‖\le ‖T‖\le ‖f_2\left(A\right)‖$$

4. Conclusions

Using the methods of convex analysis and polynomial approximation on $\left[0,+\infty \right)$, we review a simple method in evaluating the minimum of a convex function on the bounded convex subset of ${\mathbb{R}}^n.$ The first part also reviews the Bernstein theorem on integral representation of completely monotonic functions and Jensen inequality, as well as other convexity-type results. The last part of the paper comes with new consequences of previous results on Hahn–Banach-type theorems and of polynomial approximation of classes of nonnegative functions by nonnegative polynomials on the entire interval $\left[0,+\infty \right)$. Characterization for the existence and uniqueness of the solution $T$ can sometimes be made only in terms of signatures of quadratic forms (see Theorems 12 and 13). For passing to the multidimensional moment problem, it would be convenient to consider it on ${\left[0,+\infty \right)}^n, n\ge 2,$ endowed with the product measure $\mu ={\mu }_1\times \cdots \times {\mu }_n,$ where each ${\mu }_i, i=\mathrm{1,2},\dots ,n,$ is a positive regular moment determinate measure on $[0,+\infty )$ with finite moments of all positive integer orders; also, using the Fubini Theorem. The simplest example of such a measure is

$$d\mu ≔e^{-{\alpha }_1{t}_1}d{t}_1\times \cdots \times e^{-{\alpha }_n{t}_n}d{t}_n, {\alpha }_i>0, i=1,\dots ,n$$

Polynomial approximation of nonnegative functions by nonnegative polynomials on $[0,+\infty )$ represents the second key of solving the moment problems from Section 3. Similar type results can be adapted for an arbitrary unbounded closed subset $S$ of ${\mathbb{R}}^n, n\in \mathbb{N},n\ge 2$ and endowed with a moment determinate measure $\mu$ with finite moments of all orders (see [29] and the references there). The approximation holds in $L_{\mu }^1\left(S\right)$.