Generalized interpolation in H^∞ with a complexity constraint

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.