  We revisit the classical problem of absolute stability; assessing the robust
stability of a given linear time-invariant (LTI) plant in feedback with a
nonlinearity belonging to some given function class. Standard results typically
take the form of sufficient conditions on the LTI plant, the least conservative
of which are based on O'Shea--Zames--Falb multipliers. We present an
alternative analysis based on lifting and interpolation that directly
constructs a Lyapunov function that certifies absolute stability without
resorting to frequency-domain inequalities or integral quadratic constraints.
In particular, we use linear matrix inequalities to search over Lyapunov
functions that are quadratic in the iterates and linear in the corresponding
function values of the system in a lifted space. We show that our approach
recovers state-of-the-art results on a set of benchmark problems.
