设$β\equivβ^{(2n)}$是一个2n次的N维实数序列，其矩矩阵为$\mathcal{M}(n)\equiv \mathcal{M}(n)(β)$，令$r:=rank \mathcal{M}(n)$ .证明了若$\mathcal{M}(n)$是半正定的，且允许保秩矩矩阵扩张$\mathcal{M}(n+1)$，则$\mathcal{M}(n+1)$有唯一的r-原子表示测度μ，且supp μ $ equal to $ \mathcal{V}（\mathcal{M}（n+1））$, the algebraic variety of $ \mathcal{M}（n+1）$. Further, βhas an r-atomic (minimal) representing measure supported in a semi-algebraic set $ K_{ \mathcal{Q} } $ subordinate to a family $ \mathcal{Q} % \equiv \{ q_{i} \}_{i=1}^{m} \subseteq \mathbb{R} [ t_{1}，..., PDMATHABNFPDMATHABMT（n）PDMATHABNFPDMATHABMT（n+1）$ for which the associated localizing matrices $（n+[ \frac{1+°q_{i}}{2} ]）$ are positive semidefinite $（1 PDMATHABNKiPDMATHABNKm）$; in this case, μ(as above) satisfies supp μ\subseteq K_{\mathcal{Q}}$，并且μ具有精确的秩\mathcal{M}（n）-秩\mathcal{M}_{q_{i}}（n+[ \frac{1+°q_{i}}{2} ]）PDMATHABNFPDMATHABNN（q_{i}）\equiv {tPDMATHABN 0 \mathbb{R}^{N}：q_{i}（t）=0} $, $ 1 PDMATHABNKiPDMATHABNKm $。
英文摘要：
Let $β\equivβ^{(2n)}$ be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix $\mathcal{M}(n)\equiv \mathcal{M}(n)(β)$, and let $r:=rank \mathcal{M}(n)$. We prove that if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving moment matrix extension $\mathcal{M}(n+1)$, then $\mathcal{M}(n+1)$ has a unique representing measure μ, which is r-atomic, with supp μ$ equal to $\mathcal{V}(\mathcal{M}(n+1))$, the algebraic variety of $\mathcal{M}(n+1)$. Further, βhas an r-atomic (minimal) representing measure supported in a semi-algebraic set $K_{\mathcal{Q}}$ subordinate to a family $\mathcal{Q}% \equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},...,t_{N}]$ if and only if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving extension $\mathcal{M}(n+1)$ for which the associated localizing matrices $\mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$ are positive semidefinite $(1\leq i\leq m)$; in this case, μ(as above) satisfies supp μ\subseteq K_{\mathcal{Q}}$, and μhas precisely rank \mathcal{M}(n)-rank \mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$ atoms in $\mathcal{Z}(q_{i})\equiv {t\in\mathbb{R}^{N}:q_{i}(t)=0}$, $1\leq i\leq m$.
We study the truncated multidimensional moment problem with a general type of truncations. The operator approach to the moment problem is presented. A way to construct atomic solutions of the moment problem is indicated.
In this paper we study the truncated operator trigonometric moment problem. All solutions of the moment problem are described by a Nevanlinna-type parameterization. In the case of moments acting in a separable Hilbert space, the matrices of the operator coefficients in the Nevanlinna-type formula are calculated by the prescribed moments. Conditions for the determinacy of the moment problem are given, as well.
This paper is about the moment problem on a finite-dimensional vector space of continuous functions. We investigate the structure of the convex cone of moment functionals (supporting hyperplanes, exposed faces, inner points) and treat various important special topics on moment functionals (determinacy, set of atoms of representing measures, core variety).
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A constructive approach to the truncated moment problem on cubic curves in Weierstrass form
The Australian National University（澳大利亚国立大学）；University of Ljubljana, Faculty of Computer and Information Science & Faculty of Mathematics and Physics, & Institute of Mathematics, Physics and Mechanics（卢布尔雅那大学，计算机与信息科学学院与数学与物理学院，与数学、物理和力学研究所）
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Let A be a finite subset of N^n, and K be a compact semialgebraic set in R^n. An A-tms is a vector y indexed by elements in A. The A-truncated K-moment problem (A-TKMP) studies whether a given A-tms y admits a K-measure or not. This paper proposes a numerical algorithm for solving A-TKMPs. It is based on finding a flat extension of y by solving a hierarchy of semidefinite relaxations {(SDR)_k} for a moment optimization problem, whose objective R is generated in a certain randomized way. If y admits no K-measures and R[x]_A is K-full, then (SDR)_k is infeasible for all K big enough, which gives…
We consider the explicit relation between two resolvent matrices related to the truncated Hausdorff matrix moment problem (THMM) in the case of an even and odd number of moments. This relation is described with the help of four families of orthogonal matrix polynomials on the finite interval $[a,b]$ and their associated second kind polynomials.
These are ten lectures on the moment problem delivered by the author at the Vietnam Institute of Advanced Studies in Mathematics, Hanoi, March 2019. The first 3 lectures are about the one-dimensional full and truncated moment problem. The next 4 lectures deal with the multidimensional full moment problem with particular emphasis on the interactions with real algebraic geometry and polynomial optimization. The final 3 lectures are devoted to the truncated multidimensional moment problem and some recent developments in this area..
Truncated Moment Problems and the Extension Property on Monomial Curves
Indian Institute of Technology Kanpur（坎普尔印度理工学院）；University of Ljubljana, Faculty of Computer and Information Science & Faculty of Mathematics and Physics, & Institute of Mathematics, Physics and Mechanics（卢布尔雅那大学计算机与信息科学学院、数学与物理学院及数学、物理与力学研究所）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）
Let $\mathcal{A}$ be a finite-dimensional subspace of $C(\mathcal{X};\mathbb{R})$, where $\mathcal{X}$ is a locally compact Hausdorff space, and $\mathsf{A}=\{f_1,\dots,f_m\}$ a basis of $\mathcal{A}$. A sequence $s=(s_j)_{j=1}^m$ is called a moment sequence if $s_j=\int f_j(x) \, dμ(x)$, $j=1,\dots,m$, for some positive Radon measure $μ$ on $\mathcal{X}$. Each moment sequence $s$ has a finitely atomic representing measure $μ$. The smallest possible number of atoms is called the Carathéodory number $\mathcal{C}_{\mathsf{A}}(s)$. The largest number $\mathcal{C}_{\mathsf{A}}(s)$ among all moment…
In this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $\mathbb{R}^n$, and $[0,1]^n$. We also treat moment problems with small gaps. We find that for every $\varepsilon>0$ and $d\in\mathbb{N}$ there is a $n\in\mathbb{N}$ such that we can construct a moment functional $L:\mathbb{R}[x_1,\dots,x_n]_{\leq d}\rightarrow\mathbb{R}$ which needs at least $(1-\varepsilon)\cdot\left(\begin{smallmatrix} n+d\\ n\end{smallmatrix}\right)$ atoms $…
Let $\mathsf{A}=\{a_1,\dots,a_m\}$, $m\in\mathbb{N}$, be measurable functions on a measurable space $(\mathcal{X},\mathfrak{A})$. If $μ$ is a positive measure on $(\mathcal{X},\mathfrak{A})$ such that $\int a_i dμ<\infty$ for all $i$, then the sequence $(\int a_1 dμ,\dots,\int a_m dμ)$ is called a moment sequence. By Richter's Theorem each moment sequence has a $k$-atomic representing measure with $k\leq m$. The set $\mathcal{S}_\mathsf{A}$ of all moment sequences is the moment cone. The aim of this paper is to analyze the various structures of the moment cone. The main results concern the fac…
In this paper we introduce the theory of derivatives of moments and (moment) functionals to represent moment functionals by Gaussian mixtures, characteristic functions of polytopes, and simple functions of polytopes. We study, among other measures, Gaussian mixtures, their reconstruction from moments and especially the number of Gaussians needed to represent moment functionals. We find that there are moment functionals $L:\mathbb{R}[x_1,\dots,x_n]_{\leq 2d}\to\mathbb{R}$ which can be represented by a sum of $\binom{n+2d}{n} - n\cdot \binom{n+d}{n} + \binom{n}{2}$ Gaussians but not less. Hence,…
