Collaborative Research: Multivariable Moments and Factorizations and Other Problems in Analysis

合作研究：多变量矩和因式分解以及其他分析问题

• 批准号: 0500641
• 负责人: Jeffrey Geronimo
• 金额: --
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500641&HistoricalAwards=false
• 关键词: Collaborative; Research; Multivariable; Moments; Factorizations

ABSTRACTPIs Geronimo and Woerdeman will continue their investigations intofactorization and extension problems. In particular we hope todevelop a constructive proof of the celebrated Ferguson-Laceyextension theorem and build on it using the results we haverecently obtained on the spectral factorization of positive twovariable trigonometric polynomials. We plan to investigate in moredetail fast algorithms for computing the structured matrices thatarise from the two variable trigonometric moment problem as well as study the orthogonal polynomials that arise in this case. The PIsalso plan to continue their individual investigation into theimportant problems arising from quantum computing and theasymptotics of solutions to 2nd order difference equations.The types of structured matrices under study here arise in manyproblems of practical interest such as two variableauto-regressive models and two variable filtering. The PIs willtry to recruit more undergraduate and graduate students to help inthese problems as well as give courses and lectures at conferencesto increase the impact of their efforts. In order to make theintellectual merit of the proposal apparent the PIs will continueto publish their results in well respected journals, and alsodisseminate them via the PIs' homepages, preprint servers such asarXiv, software sharing websites, etc.

摘要 Geronimo 和 Woerdeman 将继续研究因式分解和扩展问题。特别是，我们希望对著名的弗格森-莱西扩张定理提出一个建设性的证明，并利用我们最近在正双变量三角多项式的谱分解上获得的结果来建立它。我们计划更详细地研究用于计算由二变量三角矩问题产生的结构化矩阵的快速算法，并研究在这种情况下出现的正交多项式。 PI还计划继续对量子计算和二阶差分方程解的渐近性产生的重要问题进行单独研究。这里研究的结构化矩阵类型出现在许多实际感兴趣的问题中，例如二变量自回归模型和二变量滤波。 PI 将尝试招募更多的本科生和研究生来帮助解决这些问题，并在会议上提供课程和讲座，以增加他们的努力的影响力。为了使该提案的智力价值显而易见，PI将继续在备受推崇的期刊上发表他们的研究结果，并通过PI的主页、预印本服务器（如sarXiv）、软件共享网站等进行传播。