New Results on Markov Moment Problem

Abstract

e present work deals with the existence of the solutions of some Markov moment problems. Necessary conditions, as well as necessary and sufficient conditions, are discussed. One recalls the background containing applications of extension results of linear operators with two constraints to the moment problem and approximation by polynomials on unbounded closed �nite-dimensional subsets. Two domain spaces are considered: spaces of absolute integrable functions and spaces of analytic functions. Operator valued moment problems are solved in the latter case. In this paper, there is a section that contains new results, making the connection to some other topics: bang-bang principle, truncated moment problem, weak compactness, and convergence. Finally, a general independent statement with respect to polynomials is discussed.

On the Moment Problem and Related Problems

“…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.…”

On Markov Moment Problem and Mazur-Orlicz Theorem

“…In [3] [4] [5] [6] [7], more results on Mazur-Orlicz theorem and the moment problem have been stated or (and) proved. The present work can be regarded as a continuation of the study from the latter works.…”

“…For example, in [11], one makes the connection to elements of fixed point theory. For the construction of a solution starting from its moments, see [6]. The same paper [6] contains connections between Markov moment problem and extreme points (Krein-Milman theorem).…”

“…For the construction of a solution starting from its moments, see [6]. The same paper [6] contains connections between Markov moment problem and extreme points (Krein-Milman theorem). In solving moment problem, three aspects are studied: the existence, the uniqueness and the construction of the solution.…”