Generalized interpolation in $H^\infty$ with a complexity constraint

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