Generalized interpolation in 𝐻∞ with a complexity constraint

In a seminal paper, Sarason generalized some classical interpolation problems for H ∞ H^∞ functions on the unit disc to problems concerning lifting onto H 2 H^2 of an operator T T that is defined on K = H 2 ⊖ ϕ H 2 K =H^2⊖ φ H^2 ( ϕ φ is an inner function) and commutes with the (compressed) shift S S . In particular, he showed that interpolants (i.e., f ∈ H ∞ f∈ H^∞ such that f ( S ) = T f(S)=T ) having norm equal to ‖ T ‖ \|T\| exist, and that in certain cases such an f f is unique and can be expressed as a fraction f = b / a f=b/a with a , b ∈ K a,b∈ K . In this paper, we study interpolants that are such fractions of