Polynomial Approximation on Unbounded Subsets, Markov Moment Problem and Other Applications

Octav Olteanu

doi:10.3390/math8101654

Abstract

This paper starts by recalling the author’s results on polynomial approximation over a Cartesian product A of closed unbounded intervals and its applications to solving Markov moment problems. Under natural assumptions, the existence and uniqueness of the solution are deduced. The characterization of the existence of the solution is formulated by two inequalities, one of which involves only quadratic forms. This is the first aim of this work. Characterizing the positivity of a bounded linear operator only by means of quadratic forms is the second aim. From the latter point of view, one solves completely the difficulty arising from the fact that there exist nonnegative polynomials on ℝn, n≥2, which are not sums of squares.

Highlights

The present paper refers mainly to various aspects of the classical multidimensional Markov moment problem and its relationship with polynomial approximation on special closed unbounded subsets A of Rn, namely on Cartesian products of closed unbounded intervals
The first aim was characterizing the existence and uniqueness of the solution for a class of Markov moment problem in terms of quadratic forms. This is only partially achieved (only one of the two inequalities appearing at point (b) referring to such characterizations involves signatures of quadratic forms)
Both subsections are based on polynomial approximation on special closed unbounded subsets

Summary

Introduction

The present paper refers mainly to various aspects of the classical multidimensional Markov moment problem and its relationship with polynomial approximation on special closed unbounded subsets A of Rn , namely on Cartesian products of closed unbounded intervals. The case n = 1 will follow as a consequence, the explicit form of nonnegative polynomials over R and, respectively, on R+ being well known (see [1]). The classical real moment problem can be formulated as follows: find necessary and sufficient conditions on a sequence y j n j∈N of real numbers, for the existence of a positive linear functional T ∈ L+ (E, R), such that. E is a Banach function space containing both polynomials as well as continuous compactly supported real-valued functions on A. It results that T can be represented by a positive regular Borel measure on

Methods

Results

Conclusion