It is argued that variational synthesis with discontinuous trial functions requires variational principles applicable to equations involving operators acting between distinct Hilbert spaces. A description is given of a Roussopoulos-type variational principle generalized to cover this situation. This principle is suggested as the basis for a unified approach to the derivation of variational functionals. In addition to esthetics, this approach has the advantage that the mathematical details increase the understanding of the derived functional, particularly the sense in which a synthesized solution should be regarded as an approximation to the true solution. By way of illustration, the generalized Roussopoulos principle is applied to derive a class of first-order diffusion functionals which admit trial functions containing approximations at an interface. These “asymptotic” interface quantities are independent of the limiting approximations from either side and permit use of different trial spectra at and on either side of an interface. The class of functionals derived contains as special cases both the Lagrange multiplier method of Buslik and two functionals of Lambropoulos and Luco. Some numerical results for a simple two-group model confirm that the “multipliers” can closely approximate the appropriate quantity in the region near an interface.