Generalized interpolation in H with a complexity constraint

Christopher I. Byrnes, Tryphon T. Georgiou, Anders Lindquist, Alexander Megretski

Research output: Contribution to journalArticlepeer-review

49 Scopus citations

Abstract

In a seminal paper, Sarason generalized some classical interpolation problems for H functions on the unit disc to problems concerning lifting onto H2 of an operator T that is defined on K = H2⊖φH2 (φ is an inner function) and commutes with the (compressed) shift 5. In particular, he showed that interpolants (i.e., f ∈ H such that f(S) = T) having norm equal to ∥T∥ exist, and that in certain cases such an / is unique and can be expressed as a fraction f = b/a with a, 6 ∈ K. In this paper, we study interpolants that are such fractions of K functions and are bounded in norm by 1 (assuming that ∥T∥ < 1, in which case they always exist). We parameterize the collection of all such pairs (a, b) ∈ K × 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 0 is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

Original languageEnglish (US)
Pages (from-to)965-987
Number of pages23
JournalTransactions of the American Mathematical Society
Volume358
Issue number3
DOIs
StatePublished - Mar 2006

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