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Generalized interpolation in H-infinity with a complexity constraint
KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.ORCID iD: 0000-0002-2681-8383
2006 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 358, no 3, 965-987 p.Article in journal (Refereed) Published
Abstract [en]

In a seminal paper, Sarason generalized some classical interpolation problems for H-infinity functions on the unit disc to problems concerning lifting onto H-2 of an operator T that is defined on K=H-2 circle minus phi H-2 (phi is an inner function) and commutes with the (compressed) shift S. In particular, he showed that interpolants (i.e., f is an element of H-infinity such that f(S)=T) having norm equal to parallel to T parallel to exist, and that in certain cases such an f is unique and can be expressed as a fraction f=b/a with a, b is an element of K. In this paper, we study interpolants that are such fractions of K functions and are bounded in norm by 1 (assuming that parallel to T parallel to<1, in which case they always exist). We parameterize the collection of all such pairs (a, b)is an element of K x 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.

Place, publisher, year, edition, pages
2006. Vol. 358, no 3, 965-987 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-37450DOI: 10.1090/S0002-9947-04-03616-5ISI: 000234197400002OAI: oai:DiVA.org:kth-37450DiVA: diva2:433900
Available from: 2011-08-11 Created: 2011-08-11 Last updated: 2017-12-08Bibliographically approved

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Lindquist, Anders

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