Konrad Schmüdgen
We prove a solvability theorem for the Stieltjes moment problem on $R^d$ which is based on the multivariate Stieltjes condition $\sum_{n=1}^\infty L(x_j^n)^{-1/(2n)}=+\infty$, $j=1,\dots,d.$ This result is applied to derive a new solvability theorem for the moment problem on unbounded semi-algebraic subsets of $R^d$.
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Multidimensional moment problem and diagonal Schur algorithm
提出一种用于多维矩问题的对角逐步 Schur 型算法，并借助 Steiltjes 变换与连分数刻画完整问题的解集，且适用于特殊截断情形。
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.
Multidimensional moment problem and Stieltjes transform
该研究利用 Stieltjes 变换把截断多维矩问题转化为插值问题，提出逐步求解算法，并用连分数刻画其全部解集。
The multidimensional truncated Moment Problem: Atoms, Determinacy, and Core Variety
本文研究连续函数在有限维向量空间上的矩问题。我们研究了凸锥的矩泛函（支撑超平面、暴露面、内点）的结构，并讨论了矩泛函的各种重要的特殊问题（确定性、表示测度的原子集、核簇）。
A Reproducing Kernel Condition for Indeterminacy in the Multidimensional Moment Problem
利用与多维矩问题相关的Hankel形式的最小特征值，我们建立了一个等价于再生核存在的条件。这一结果是Berg、Chen和Ismail 2002年结果的多变量模拟。我们还提出了一类措施，再生核的存在意味着不确定性。
We consider the problem of finding a (non-negative) measure $μ$ on $\mathfrak{B}(\mathbb{C}^n)$ such that $\int_{\mathbb{C}^n} \mathbf{z}^{\mathbf{k}} dμ(\mathbf{z}) = s_{\mathbf{k}}$, $\forall \mathbf{k}\in\mathcal{K}$. Here $\mathcal{K}$ is an arbitrary finite subset of $\mathbb{Z}^n_+$, which contains $(0,...,0)$, and $s_{\mathbf{k}}$ are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. At first, one may consider this problem as an extension of the truncated multidimensional moment problem on $\mathbb{R}^n$, w…
We will consider the indefinite truncated multidimensional moment problem. Necessary and sufficient conditions for a given truncated multisequence to have a signed representing measure $μ$ with ${\rm card}\,{\rm supp}\, μ$ as small as possible are given by the existence of a rank preserving extension of a multivariate Hankel matrix (built from the given truncated multisequence) such that the corresponding associated polynomial ideal is real radical. This result is a special case of a more general characterisation of truncated multisequences with a minimal complex representing measure whose sup…
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..
The multidimensional truncated Moment Problem: Carathéodory Numbers
设$\mathcal{A}$是$C(\mathcal{X};\mathbb{R})$的有限维子空间，其中$\mathcal{X}$是局部紧Hausdorff空间，$\mathsf{A}=\{f_1,\dots,f_m\}$是$\mathcal{A}$的基.一个序列$s=(s_j)_{j=1}^m$称为矩序列，如果对于某个正氡测度$μ$在$\mathcal{X}$上，有$s_j=\int f_j(x) \, dμ(x)$，$j=1,\dots,m$。每个矩序列$s$具有一个表示测度$μ$的原子。最小的原子数称为Carathéodory数$\mathcal{C}_{\mathsf{A}}(s)$。所有矩序列$s$中的最大数$\mathcal{C}_{\mathsf{A}}(s)$是Carathéodory数$\mathcal{C}_{\mathsf{A}}$。本文研究了Carathéodory数$\mathcal{C}_{\mathsf{A}}(s)$和$\mathcal{C}_{\mathsf{A}}$.在可微函数的情况下，使用微分几何的方法。重点是实多项式。对于一个大类的空间多项式在一个变量的数量$\mathcal{C}_{\mathsf{A}}$确定。在多变量的情况下，我们得到一些下界，我们使用的正多项式的零上的结果推导出Carathéodory数的上界。
Generalized Widder Theorem via fractional moments
当表示测度的支集包含在某个广义半代数集中时，给出了函数可表示为经典多维拉普拉斯变换的一个充要条件.这是通过采用Putinar和Vasilescu [Putinar，M.和Vasilescu，F.- H、通过维数扩展求解矩问题，Ann. of Math.（2）149（1999），no.3，1087-1107]，用于相应的多维矩问题。
On the Real Multidimensional Rational K-Moment Problem
本文给出了实多维有理K-矩问题的一种解法，其中K由多个多项式不等式定义。更确切地说，设S是X=（X_1，.，X_n），使得相应的基本闭半代数集KS非空。设E=D^{-1}R[X]是实多项式代数的局部化，T_S^E是E上由S生成的预序.我们证明了E上的每个在T_S^E上非负的线性泛函L都由K_S的某个子集上的正测度表示，只要D包含在K_S上增长足够快的元素.
The Multidimensional Moment Problem with Complexity Constraint
以前的一系列论文都致力于研究具有非负有理数测度的（一维）矩问题。合理性假设是一个复杂的约束，其中的应用程序的参数化的解决方案集的有限数量的参数是必需的。在本文中，我们提供了一个完整的解决方案的多维矩问题的复杂性约束，也允许解决方案，需要一个奇异的措施添加到合理的，绝对连续的。这样的解出现在解的某个凸锥的边界上。在本文中，我们提供了完整的参数化所有这样的解决方案。我们还提供了由本论文的作者之一合著的该杂志上的一篇论文的勘误表。
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 our companion paper "Multidimensional rational covariance extension with applications to spectral estimation and image compression" we discussed the multidimensional rational covariance extension problem (RCEP), which has important applications in image processing, and spectral estimation in radar, sonar, and medical imaging. This is an inverse problem where a power spectrum with a rational absolutely continuous part is reconstructed from a finite set of moments. However, in most applications these moments are determined from observed data and are therefore only approximate, and RCEP may no…
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,…
Multidimensional Rational Covariance Extension with Applications to Spectral Estimation and Image Compression
有理协方差延拓问题（RCEP）是系统与控制中的一个重要问题，涉及控制、估计、系统辨识、信号与图像处理等多个领域，涉及许多基本理论问题。事实上，这个逆问题是许多识别和信号处理技术的关键组成部分，并在系统和信号的预测，分析和建模中发挥着重要作用。众所周知，RCEP可以重新表述为一个（截断）三角矩问题，并满足合理性条件。在本文中，我们考虑更一般的多维三角矩问题具有类似的合理性约束。这种推广创造了许多有趣的新的数学问题，也为原来的一维问题提供了新的见解。这种方法的一个关键概念是所有解决方案的完整平滑参数化，允许调整解决方案以满足额外的设计规范，而不违反复杂性约束。作为这种方法的潜力的说明，我们将我们的结果应用到多维谱估计和图像压缩。这仅仅是朝着这个方向迈出的第一步，我们希望未来更详细的调优策略将增强我们的程序。
Identification of the theory of multidimensional orthogonal polynomials with the theory of symmetric interacting Fock spaces with finite dimensional one particle space
在标题中提到的标识允许制定的多维Favard引理不同于目前在文献中使用的，它完全平行于原来的一维配方的意义上，正雅可比序列被替换为一个序列的正厄米（平方）矩阵和真正的雅可比序列由一个序列的厄米矩阵相同的尺寸。此外，在这种识别中，正Jacobi序列的相容性条件的多维扩展成为保证相互作用的Fock空间中的创造者存在的条件。上述结果为任意阶矩有限的$\mathbb{R}^d$上概率测度的纯代数分类的程序化开辟了道路。在这个分类中，$\mathbb{C}^d$上的Boson-Fock空间的特征是正Jacobi序列由单位矩阵组成，实Jacobi序列恒零.经典实值随机变量的全矩量子分解是证明的主要内容之一。
Multidimensional Chebyshev spaces, hierarchy of infinite-dimensional spaces and Kolmogorov-Gelfand widths
本文引入基于高阶椭圆方程解的多维切比雪夫空间，建立无限维空间层级，并计算多维域上特殊椭球函数集的宽度。
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