On Markov Moment Problem, Polynomial Approximation on Unbounded Subsets, and Mazur–Orlicz Theorem

Octav Olteanu

doi:10.3390/sym13101967

Abstract

We review earlier and recent results on the Markov moment problem and related polynomial approximation on unbounded subsets. Such results allow proving the existence and uniqueness of the solutions for some Markov moment problems. This is the first aim of the paper. Our solutions have a codomain space a commutative algebra of (linear) symmetric operators acting from the entire real or complex Hilbert space H to H; this algebra of operators is also an order complete Banach lattice. In particular, Hahn–Banach type theorems for the extension of linear operators having a codomain such a space can be applied. The truncated moment problem is briefly discussed by means of reference citations. This is the second purpose of the paper. In the end, a general extension theorem for linear operators with two constraints is recalled and applied to concrete spaces. Here polynomial approximation plays no role. This is the third aim of this work.

Highlights

The moment problem was formulated by T
The numbers y j, j ∈ Nn are called the moments of the measure μ
Extension of linear operators, satisfying a sandwich condition. Such results are used in the existence of a solution for some Markov moment problems and the Mazur–Orlicz theorem

Summary

Introduction

The moment problem was formulated by T. The existence, uniqueness, and eventually the construction of the solution dσ starting from its moments 0 tk dσ, k ∈ N is under attention. The problems stated above have been generalized as follows: being given a sequence y j j∈Nn of real numbers and a closed subset F ⊆ Rn , n ∈ {1, 2, . . .}, find a positive regular Borel measure μ on F such that F t j dμ = y j , j ∈ Nn. The problems stated above have been generalized as follows: being given a sequence y j j∈Nn of real numbers and a closed subset F ⊆ Rn , n ∈ {1, 2, . . .}, find a positive regular Borel measure μ on F such that F t j dμ = y j , j ∈ Nn The direct problem could be: being given the measure μ, find its moments

Methods

Results

Conclusion