On Markov Moment Problem and Related Results

Octav Olteanu

doi:10.3390/sym13060986

Abstract

We prove new results and complete our recently published theorems on the vector-valued Markov moment problem, by means of polynomial approximation on unbounded subsets, also applying an extension of the positive linear operators’ result. The domain is the Banach lattice of continuous real-valued functions on a compact subset or an Lν1 space, where ν is a positive moment determinate measure on a closed unbounded set. The existence and uniqueness of the operator solution are proved. Our solutions satisfy the interpolation moment conditions and are between two given linear operators on the positive cone of the domain space. The norm controlling of the solution is emphasized. The most part of the results are stated and proved in terms of quadratic forms. This type of result represents the first aim of the paper. Secondly, we construct a polynomial solution for a truncated multidimensional moment problem.

Highlights

The one dimensional classical moment can be formulated as follows: for a given sequencen≥0 of real numbers, find a positive measure ν on the given interval I or I = [0, 1]), such that the following applies: tndν = yn, n = 0, 1, 2, . . . (1) IReceived: 22 April 2021 Accepted: 25 May 2021 Published: 1 June 2021 where the numbers yn, n = 0, 1, 2, . . . are called the moments of the measure dν on the interval I
We prove new results and complete our recently published theorems on the vector-valued Markov moment problem, by means of polynomial approximation on unbounded subsets, applying an extension of the positive linear operators’ result
We construct a polynomial solution for a truncated multidimensional moment problem

Summary

Introduction

The one dimensional classical moment can be formulated as follows: for a given sequence (yn)n≥0 of real numbers, find a positive measure ν on the given interval I (the important cases being I = R, I = [0, ∞) or I = [0, 1]), such that the following applies: tndν = yn, n = 0, 1, 2, . Are called the moments of the measure dν (or ν) on the interval I. The function is defined as follows: ( f1 ⊗ · · · ⊗ fn)(t1, . An important case is when the solution is a positive linear functional or operator defined on L1ν(S). In this case, ν is assumed to be a positive regular Borel measure on S. A basic particular case is that of a determinate measure ν, with finite moments of all orders: tjdν ∈ R, j ∈ Nn

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Conclusion