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

Geronimo和Woerdeman将继续研究因式分解和扩展问题。特别地，我们希望给出著名的Ferguson-Lacey延拓定理的一个构造性证明，并利用我们最近得到的关于正二元三角多项式的谱分解的结果来建立它。我们计划更详细地研究计算由两个变量三角矩问题产生的结构矩阵的快速算法，以及在这种情况下产生的正交多项式。他们还计划继续对量子计算和二阶差分方程解的渐近性等重要问题进行个人研究。这里研究的结构矩阵类型出现在许多实际感兴趣的问题中，如两个变量自回归模型和两个变量过滤。PIs将努力招募更多的本科生和研究生来帮助解决这些问题，并在会议上开设课程和讲座，以增加他们的努力的影响。为了使该建议的学术价值显而易见，私人投资机构将继续在备受尊敬的期刊上发表其成果，并通过个人投资机构的主页、预印服务器(如ARXIV)、软件共享网站等进行传播。

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.