From Hahn–Banach Type Theorems to the Markov Moment Problem, Sandwich Theorems and Further Applications

Abstract

The aim of this review paper is to recall known solutions for two Markov moment problems, which can be formulated as Hahn–Banach extension theorems, in order to emphasize their relationship with the following problems: (1) pointing out a previously published sandwich theorem of the type f ≤ h ≤ g, where f, −g are convex functionals and h is an affine functional, over a finite-simplicial set X, and proving a topological version for this result; (2) characterizing isotonicity of convex operators over arbitrary convex cones; giving a sharp direct proof for one of the generalizations of Hahn–Banach theorem applied to the isotonicity; (3) extending inequalities assumed to be valid on a small subset, to the entire positive cone of the domain space, via Krein–Milman or Carathéodory’s theorem. Thus, we point out some earlier, as well as new applications of the Hahn–Banach type theorems, emphasizing the topological versions of these applications.

On the Moment Problem and Related Problems

“…It is worth noting that Theorem 2 leads to a sandwich-type theorem on the existence of an affine functional h, f ≤ h ≤ g on S, where S is a finite-simplicial (convex) subset of a vector space, while f and -g are convex functionals such that f ≤ g on S. The novelty is that a finite-simplicial set might be unbounded in any locally convex topology on the domain space containing S (see [40,41]). A topological version has been published in [41], Theorem 4. Theorem 3.…”

“…Theorem 3. (see [35], Theorem 1 and [41], Theorem 5). Let X be a preordered vector space, Y an order complete vector lattice, P : X → Y a convex operator, and x j j∈J ⊂ X, y j j∈J ⊂ Y given families.…”

“…Various aspects of the moment problem are discussed in [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. For other applications, results on Markov moment problem and connections with other fields of analysis see [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. The moment problem has applications in physics, probability theory, and statistics, as discussed in the Introduction of [2].…”

“…The paper [29] contains the first example of a non-negative polynomial on R 2 that is not a sum of squares. The papers [30,31] emphasize optimization problems related to the moment problem, while in [32][33][34][35][36][37][38][39][40][41][42] various aspects of the full and of the truncated Markov moment problem are examined. All of the main results of the present work concerning the Markov moment problem are valid for the vector-valued problem.…”

“…From this point of view, the difficulty arising from the fact that non-negative polynomials on R n , n ≥ 2 are not sums of squares is solved. Another application of the solution to an abstract moment problem (Theorem 2, stated below) to an unexpected sandwich result is briefly reviewed with the aid of the referenced citations (see [40,41], Theorems 2 and 4).…”