Generalized interpolation in H-infinity with a complexity constraint
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
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.
IdentifiersURN: urn:nbn:se:kth:diva-37450DOI: 10.1090/S0002-9947-04-03616-5ISI: 000234197400002OAI: oai:DiVA.org:kth-37450DiVA: diva2:433900