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.
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.
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.
The aim of this review paper is to recall known solutions for two Markov moment problems, which can be formulated as Hahn–Banach extension theorems, in order to emphasize their relationship with the following problems: (1) pointing out a previously published sandwich theorem of the type f ≤ h ≤ g, where f, −g are convex functionals and h is an affine functional, over a finite-simplicial set X, and proving a topological version for this result; (2) characterizing isotonicity of convex operators over arbitrary convex cones; giving a sharp direct proof for one of the generalizations of Hahn–Banach theorem applied to the isotonicity; (3) extending inequalities assumed to be valid on a small subset, to the entire positive cone of the domain space, via Krein–Milman or Carathéodory’s theorem. Thus, we point out some earlier, as well as new applications of the Hahn–Banach type theorems, emphasizing the topological versions of these applications.
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