New Results on Markov Moment Problem

Octav Olteanu

doi:10.1155/2013/901318

Abstract

The 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 finite-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.

Highlights

Introduction and Known ResultsWe recall that the classical formulation of the moment problem, under the terms of T
Using a quite different proof with respect to that of eorem 9, we proved the following “abstract operatorial version.”
Let FF2 ∶ LL1νν(RRR R RR be a linear continuous positive operator vanishing on the subspace of odd functions and {yy2kk}kk ⊂ YY

Summary

Octav Olteanu

E present work deals with the existence of the solutions of some Markov moment problems. 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. 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. A general independent statement with respect to polynomials is discussed

Introduction and Known Results

One obtains

AAkk ρρkk

Further computation yields

TT xxjj h

We also infer that yyjj