Colloquia Patavina: The moment problem for positive rational measures: convexity in the spirit of Krein


Martedi' 13 Ottobre 2009 alle ore 16:00 in aula 1A/150 della Torre Archimede il Prof. Anders Lindquist (Royal Institute of Technology) terra' una conferenza della serie Colloquia Patavina.

La Commissione Colloquia
M.A. Garuti, M. Pavon, M. Pitteri, F. Rossi

The moment problem for positive rational measures: convexity in the
spirit of Krein

Anders Lindquist (Royal Institute of Technology)

The moment problem as formulated by Krein and Nudel'man is a beautiful generalization of several important classical moment problems, including the power moment problem, the trigonometric moment problem and the moment problem arising in Nevanlinna-Pick interpolation. However, the importance of rational functions in systems and control and other engineering applications imposes certain complexity constraints. In this talk we present a new formulation of the moment problem which respects these constraints. While this version of the problem is decidedly nonlinear, the basic tools still rely on convexity. We give a complete parameterization of all solutions. This can be seen as a global analysis approach, where one studies an entire class of solutions as a whole. We then show that each solution in this class can be obtained by minimizing a strictly convex nonlinear functional. Thus the methodology employed is a combination of nonlinear analysis, geometry and optimization. Finally, we apply these results to interpolation problems of the Caratheodory and of the Nevanlinna-Pick type, arising in signal processing and control theory, where we consider smooth bijective transformations from spaces of tuning parameters to entire classes of solutions.

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