Abstract
Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence (yj)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 ∫Ftjdμ=yj, j∈ℕn. This is the full moment problem. The existence, uniqueness, and construction of the unknown solution μ are the focus of attention. The numbers yj, j∈ℕn are called the moments of the measure μ. When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments yj are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work.
Highlights
The moment problem is recalled in the Abstract
The polynomial approximation results have been motivated by the Markov moment problem, but they led to characterization of the positivity of some bounded linear operators only in terms of quadratic forms, even in the case of spaces of functions of several variables
Sufficient criteria for determinacy and indeterminacy are under attention; 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]
Summary
The moment problem is recalled in the Abstract. Originally, it was formulated by T.Stieltjes in 1894–1895 (see [1]): find the repartition of the positive mass on the non-negative semi-axis, if the moments of arbitrary orders k (k = 0, 1, 2, . . .) are given. The moment problem is recalled in the Abstract. Stieltjes in 1894–1895 (see [1]): find the repartition of the positive mass on the non-negative semi-axis, if the moments of arbitrary orders k In the Stieltjes moment problem, a sequence of real numbers (yk )k≥0 is given, and one looks for a R∞. Nondecreasing real function σ (t) (t ≥ 0), which verifies the moment conditions: 0 tk dσ =. If such a function σ does exist, the sequence (yk )k≥0 is called a Stieltjes moment sequence. A Hamburger moment sequence is a sequence (yk )k≥0 for which there
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