In this paper, we consider input-output properties of linear systems consisting of PDEs on a finite domain coupled with ODEs through the boundary conditions of the PDE. This work generalizes the sufficiency proof of the KYP Lemma for ODEs to coupled ODE-PDE systems using a recently developed concept of fundamental state and the associated boundary-condition-free representation. The conditions of the generalized KYP are tested using a positive matrix parameterization of bounded operators resulting in a finite-dimensional LMI, the feasibility of which implies prima facie provable passivity or Lgain of the system. No discretization or approximation is involved at any step and there is no conservatism in the theorems. Comparison with other computational methods show that bounds obtained are not conservative in any significant sense and that computational complexity is lower than existing methods involving finite-dimensional projection of PDEs.