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

Geronimo和Woerdeman将继续他们对因子分解和可拓问题的研究。特别是，我们希望发展一个建设性的证明著名的弗格森-莱西扩展定理，并建立在它使用的结果，我们最近得到的频谱因式分解的正二元三角多项式.我们计划调查更详细的快速算法计算结构矩阵thatarise从两个变量的三角矩问题，以及研究正交多项式，在这种情况下出现. PI还计划继续对量子计算和二阶差分方程解的渐近性所产生的重要问题进行单独研究。这里研究的结构矩阵类型出现在许多实际问题中，例如双变量自回归模型和双变量滤波。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，并在会议上开设课程和讲座，以增加他们的努力的影响。为了使该提案的学术价值显而易见，PI将继续在备受尊敬的期刊上发表他们的结果，并通过PI的主页，预印本服务器（如arXiv），软件共享网站等传播它们。

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.