For a finite positive Borel measure μ on the unit circle, let D(μ) be the associated harmonically weighted Dirichlet space. A shift invariant subspace M recognizes strong approximate spectral cosynthesis if there exists a sequence of shift invariant subspaces Mk, with finite codimension, such that the orthogonal projections onto Mk converge in the strong operator topology to the orthogonal projection onto M. In this talk, we discuss the result that proves that shift invariant subspaces of D(μ) admit strong approximate spectral cosynthesis whenever μ is a finite sum of atoms.