Solving moment problems by dimensional extension

2017

This thesis contains an exposition of the Hamburger moment problem. The Hamburger moment problem is an interesting question in analysis that deals with finding the existence of a Borel measure representing a given positive semi-definite linear functional. We begin our exposition by constructing orthogonal polynomials associated with a positive definite sequence. Then we discuss the interlacing property of the zeros of these orthogonal polynomials. We proceed by finding a solution to the truncated Hamburger moment problem and then extend the found solution to the complete Hamburger moment problem. After obtaining a solution to the Hamburger moment problem, we address the problem of determinacy of the moment problem. Finally, we discuss a result that proves the density of polynomials with complex coefficients under the assumption that the Carleman’s condition is satisfied.

Revue Roumaine de Mathematiques …, 2007

We consider a class of positive polynomials on the whole interval [0, ∞[, related to the functions exp(−kt), t ∈ [0, ∞[, k ∈ Z+. We prove that for any positive Borel regular measure ν on [0, ∞[ with moments of all orders, these polynomials are dense in (L 1 ν ([0, ∞[))+. We solve a determinate Markov-type moment problem on [0, ∞[. The Markov moment problem on arbitrary compacts K ⊂ R n is reduced to the Markov problem for semi-algebraic compacts KA ⊂ R n . The "non-compact case" is reduced to the "compact case" for some Borel subsets of R n .

Journal of Computational and Applied Mathematics, 1993

Torrano, E. and R. Guadalupe, On the moment problem in the bounded case, Journal of Computational and Applied Mathematics 49 Through the matrix treatment of the theory of orthogonal polynomials on curves or domains of the complex plane, we extend to arbitrary bounded regions the results of for the unit disc.

arXiv (Cornell University), 2023

We summarize significant classical results on (in)determinacy of measures in terms of their finite positive integer order moments. Well-known is the role of the smallest eigenvalues of Hankel matrices, starting from Hamburger's results a century ago and ending with the great progress made only in recent times by C. Berg and collaborators. We describe here known results containing necessary and sufficient conditions for moment (in)determinacy in both Hamburger and Stieltjes moment problems. In our exposition we follow an approach different from that commonly used. There are novelties well complementing the existing theory. Among them are: (a) to emphasize on the geometric interpretation of the indeterminacy conditions; (b) exploit fine properties of the eigenvalues of perturbed symmetric matrices allowing to derive new lower bounds for the smallest eigenvalues of Hankel matrices; these bounds are used for concluding indeterminacy; (c) provide new arguments to confirm classical results; (d) give new numerical illustrations involving commonly used probability distributions.

Journal of Functional Analysis, 2008

Let β ≡ β (2n) = {βi} |i|≤2n denote a d-dimensional real multisequence, let K denote a closed subset of R d , and let P2n := {p ∈ R[x1, . . . , x d ] : deg p ≤ 2n}. Corresponding to β, the Riesz functional L ≡ L β : P2n −→ R is defined by L( aix i ) := aiβi. We say that L is K-positive if whenever p ∈ P2n and p|K ≥ 0, then L(p) ≥ 0. We prove that β admits a K-representing measure if and only if L β admits a K-positive linear extensionL : P2n+2 −→ R. This provides a generalization (from the full moment problem to the truncated moment problem) of the Riesz-Haviland Theorem. We also show that a semialgebraic set solves the truncated moment problem in terms of natural "degree-bounded" positivity conditions if and only if each polynomial strictly positive on that set admits a degree-bounded weighted sum-of-squares representation.

Bulletin des Sciences Mathématiques, 2011

We consider the problem of vanishing of the moments m k (P , q) = Ω P k (x)q(x) dμ(x) = 0, k = 1, 2,. .. , with Ω a compact domain in R n and P (x), q(x) complex polynomials in x ∈ Ω (MVP). The main stress is on relations of this general vanishing problem to the following conjecture which has been studied recently in Mathieu (1997) [22], Duistermaat and van der Kallen (1998) [17], Zhao (2010) [34,35] and in other publications in connection with the vanishing problem for differential operators and with the Jacobian conjecture: Conjecture A. For positive μ if m k (P , 1) = 0 for k = 1, 2,. .. , then m k (P , q) = 0 for k 1 for any q. We recall recent results on one-dimensional (MVP) obtained in Muzychuk and Pakovich (2009) [24], Pakovich (2009,2004) [25,26], Pakovich (preprint) [28] and prove some initial results in several variables, stressing the role of the positivity assumption on the measure μ. On this base we analyze some special cases of Conjecture A and provide in these cases a complete characterization of the measures μ for which this conjecture holds.

Integral Equations and Operator Theory, 2008

For a degree 2n real d-dimensional multisequence β ≡ β (2n) = {β i } i∈Z d + ,|i|≤2n to have a representing measure µ, it is necessary for the associated moment matrix M(n)(β) to be positive semidefinite and for the algebraic variety associated to β, V ≡ V β , to satisfy rank M(n) ≤ card V as well as the following consistency condition: if a polynomial p(x) ≡ |i|≤2n a i x i vanishes on V, then |i|≤2n a i β i = 0. We prove that for the extremal case (rank M(n) = card V), positivity of M(n) and consistency are sufficient for the existence of a (unique, rank M(n)-atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of M(n).

Let $\ast_P$ be a product on $l_{\rm{fin}}$ (a space of all finite sequences) associated with a fixed family $(P_n)_{n=0}^{\infty}$ of real polynomials on $\mathbb{R}$. In this article, using methods from the theory of generalized eigenvector expansion, we investigate moment-type properties of $\ast_P$-positive functionals on $l_{\rm{fin}}$. If $(P_n)_{n=0}^{\infty}$ is a family of the Newton polynomials $P_n(x)=\prod_{i=0}^{n-1}(x-i)$ then the corresponding product $\star=\ast_P$ is an analog of the so-called Kondratiev--Kuna convolution on a ``Fock space''. We get an explicit expression for the product $\star$ and establish a connection between $\star$-positive functionals on $l_{\rm{fin}}$ and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).

Reports on Mathematical Physics, 2000

2010

This note is devoted to the L-moment problem. The L-moment problem consists of characterising the sequence of moments an = ∫ Rt nf(t)dt, n ∈ N of a real measurable function f (with prescribed support) which satisfies a condition such is 0 ≤ f ≤ L a.e. dt. The L-moment problem was formulated and completely solved by Akhiezer and Krein [2] in the thirties. The interest for the moments of a bounded measurable function on the real axis goes back to A.A. Markov in the last part of the nineteenth century. It was M.G. Krein who studied again this field with new methods at that time of function theory and functional analysis. The aim of the present paper is to study a two dimensional complex L-moment problem. The L-complex moment problem consists of characterising a double complex sequence {yα,β}α,β∈N to represent the moments of a measurable real function h defined on the closed unit disc of the complex plane, which satisfies a condition of boundness 0 ≤ h ≤ L for a positive constant L >...