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

Octav Olteanu

doi:10.3390/math8081328

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.

Highlights

We recall the classical formulation of the moment problem, under the terms of T
The above results point out four directions for applications of generalizations of Hahn–Banach theorem, mentioned in the end of the Introduction
As we have already seen in the Introduction, solving multidimensional moment problems in terms of signatures of quadratic forms is a difficult task, since nonnegative polynomials on closed subsets of Rn, n ≥ 2 generally are not expressible by means of sums of squares

Summary

Introduction

We recall the classical formulation of the moment problem, under the terms of T. Given in 1894–1895 (see the basic book of N.I. Akhiezer [1] for details): find the repartition of the positive mass on the nonnegative semi-axis, if the moments of arbitrary orders k In the Stieltjes moment problem, a sequence of real numbers (Sk )k≥0 is given and one looks for a nondecreasing real function σ(t) (t ≥ 0), which verifies the moment conditions: Z∞. This is a one dimensional moment problem, on an unbounded interval. It is an interpolation problem with the constraint on the positivity of the measure dσ.

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Conclusion