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

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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.