On Markov Moment Problem and Related Results

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 space, where 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.

On the Moment Problem and Related Problems

“…These methods can be applied to the Markov moment problem as well (see [35], Theorem 2 and [2], Corollary 12.29). Sufficient criteria for determinacy and indeterminacy are also under attention (see [2,20]); (3) We review our earlier results on polynomial approximation on unbounded subsets, and its applications to the vector-valued Markov moment problem, recently published, completed, and generalized in [38] (see also [36]). To this aim, we essentially use the notion of a moment-determinate (M− determinate) measure (see [2,13,20]); (4) Two results on truncated scalar-valued Markov moment problem [37] are briefly discussed.…”

“…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.…”

“…The case of the corresponding scalar-valued problem follows as a consequence. Although the full moment problem is discussed in detail in Section 3 (because it could have a unique solution, whose existence is characterized in terms of quadratic forms (see [36][37][38]), the truncated moment problem is also one of the aims of the same section. Major results on this subject have been published in [24,25,[30][31][32][33][37][38][39].…”

“…Although the full moment problem is discussed in detail in Section 3 (because it could have a unique solution, whose existence is characterized in terms of quadratic forms (see [36][37][38]), the truncated moment problem is also one of the aims of the same section. Major results on this subject have been published in [24,25,[30][31][32][33][37][38][39]. In the end, as an application of polynomial approximation in another direction, the positivity of a bounded linear operator is also characterized in terms of quadratic forms (see [36], Theorem 7, and Theorem 18 proved in Section 3.2 of the present work).…”