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Generalized interpolation in  with a complexity constraint |
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| Authors: | Christopher I. Byrnes Tryphon T. Georgiou Anders Lindquist Alexander Megretski |
| Affiliation: | Department of Electrical and Systems Engineering, Washington University, St. Louis, Missouri 63130 Tryphon T. Georgiou ; Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota 55455 Anders Lindquist ; Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden Alexander Megretski ; Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307 |
| Abstract: | In a seminal paper, Sarason generalized some classical interpolation problems for  functions on the unit disc to problems concerning lifting onto  of an operator  that is defined on  ( is an inner function) and commutes with the (compressed) shift  . In particular, he showed that interpolants (i.e.,  such that  ) having norm equal to  exist, and that in certain cases such an  is unique and can be expressed as a fraction  with  . In this paper, we study interpolants that are such fractions of  functions and are bounded in norm by  (assuming that  , in which case they always exist). We parameterize the collection of all such pairs  and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where  is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint. |
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