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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Generalized interpolation in $H^\infty$ with a complexity constraint


Authors: Christopher I. Byrnes, Tryphon T. Georgiou, Anders Lindquist and Alexander Megretski
Journal: Trans. Amer. Math. Soc. 358 (2006), 965-987
MSC (2000): Primary 47A57, 30E05; Secondary 46N10, 47N10, 93B15
Published electronically: December 9, 2004
MathSciNet review: 2187641
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Abstract: In a seminal paper, Sarason generalized some classical interpolation problems for $H^\infty$ functions on the unit disc to problems concerning lifting onto $H^2$ of an operator $T$ that is defined on $\EuScript{K} =H^2\ominus\phi H^2$($\phi$ is an inner function) and commutes with the (compressed) shift $S$. In particular, he showed that interpolants (i.e., $f\in H^\infty$ such that $f(S)=T$) having norm equal to $\Vert T\Vert$ exist, and that in certain cases such an $f$ is unique and can be expressed as a fraction $f=b/a$ with $a,b\in\EuScript{K}$. In this paper, we study interpolants that are such fractions of $\EuScript{K}$ functions and are bounded in norm by $1$ (assuming that $\Vert T\Vert<1$, in which case they always exist). We parameterize the collection of all such pairs $(a,b)\in\EuScript{K}\times\EuScript{K}$ 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 $\phi$ is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.


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Additional Information

Christopher I. Byrnes
Affiliation: Department of Electrical and Systems Engineering, Washington University, St. Louis, Missouri 63130

Tryphon T. Georgiou
Affiliation: Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota 55455

Anders Lindquist
Affiliation: Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden

Alexander Megretski
Affiliation: Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03616-5
PII: S 0002-9947(04)03616-5
Received by editor(s): October 27, 2003
Received by editor(s) in revised form: January 21, 2004
Published electronically: December 9, 2004
Additional Notes: This research was supported in part by Institut Mittag-Leffler and by grants from AFOSR, NSF, VR, the Göran Gustafsson Foundation, and Southwestern Bell.
Article copyright: © Copyright 2004 American Mathematical Society