In a seminal paper, Sarason generalized some classical interpolation problems for
H
∞
H^\infty
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
\mathcal {K} =H^2\ominus \phi H^2
(
ϕ
\phi
is an inner function) and commutes with the (compressed) shift
S
S
. In particular, he showed that interpolants (i.e.,
f
∈
H
∞
f\in H^\infty
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\in \mathcal {K}
. In this paper, we study interpolants that are such fractions of
K
\mathcal {K}
functions and are bounded in norm by
1
1
(assuming that
‖
T
‖
>
1
\|T\|>1
, in which case they always exist). We parameterize the collection of all such pairs
(
a
,
b
)
∈
K
×
K
(a,b)\in \mathcal {K}\times \mathcal {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
ϕ
\phi
is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.