多元矩问题的再思考与多元积分逼近的新方法
Reconsideration of the multivariate moment problem and a new method for approximating multivariate integrals
AI总结：
本文通过引入伪正测度替代正测度，解决了多元矩问题，并基于Gauss-Jacobi过程的推广构造了最小支撑的伪正测度，从而发展了高效的多元积分求积公式。
摘要：
由于与谱理论和薛定谔算子的密切关系，多元矩问题一直是许多研究的主题，但迄今为止尚未取得实质性成功（与一维情况相比）。在本文中，我们重新考虑了标准方法的一个基本公理——测度的正性。我们引入了所谓的伪正测度。我们的主要成就之一是在伪正测度类中解决了矩问题。测度 $μ$ 被称为伪正测度，如果其在球谐展开中的Laplace-Fourier系数 $μ_{k,l}(r)$，$r\geq0$，是非负的。我们方法的另一个主要优点是，对于伪正测度，我们可以通过推广经典的Gauss-Jacobi过程来发展有效的“求积公式”：对于每个整数 $p\geq1$，我们构造一个新的伪正测度 $ν_{p}$，它具有“最小支撑”，并且对于所有满足 $Δ^{2p}h=0$ 的多项式 $h$，有 $μ(h)=ν_{p}(h)$。这一结果的证明需要应用Chebyshev、Markov、Stieltjes、Krein关于Gauss-Jacobi测度极值性质的著名理论，通过使用关于每个测度 $μ_{k,l}$ 的经典正交多项式 $p_{k,l;j}$，$j\geq0$。作为副产品，我们得到了由多项式 $p_{k,l;j}$ 定义的多元正交性的概念。我们研究的一个主要动机是进一步开发多元薛定谔算子的新模型，这推广了M. Stone的经典结果，即一维正交多项式代表了具有简单谱的自伴算子的一个模型。
英文摘要：
Due to its intimate relation to Spectral Theory and Schrödinger operators, the multivariate moment problem has been a subject of many researches, so far without essential success (if one compares with the one--dimensional case). In the present paper we reconsider a basic axiom of the standard approach - the positivity of the measure. We introduce the so--called pseudopositive measures instead. One of our main achievements is the solution of the moment problem in the class of the pseudopositive measures. A measure \ $μ$ is called pseudopositive if its Laplace-Fourier coefficients $μ_{k,l}(r) ,$ $r\geq0,$ in the expansion in spherical harmonics are non--negative. Another main profit of our approach is that for pseudopositive measures we may develop efficient ''cubature formulas'' by generalizing the classical procedure of Gauss--Jacobi: for every integer \ $p\geq1$ we construct a new pseudopositive measure $ν_{p}$ having ''minimal support'' and such that $μ(h) =ν_{p}(h) $ for every polynomial $h$ with $Δ^{2p}h=0.$ The proof of this result requires application of the famous theory of Chebyshev, Markov, Stieltjes, Krein for extremal properties of the Gauss-Jacobi measure, by employing the classical orthogonal polynomials $p_{k,l;j},$ $j\geq0,$ with respect to every measure $μ_{k,l}.$ As a byproduct we obtain a notion of multivariate orthogonality defined by the polynomials $p_{k,l;j}$. A major motivation for our investigation has been the further development of new models for the multivariate Schrödinger operators, which generalize the classical result of M. Stone saying that the one--dimensional orthogonal polynomials represent a model for the self--adjoint operators with simple spectrum.
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