General Optimal Polynomial Approximants, Stabilization, and Projections of Unity

In various Hilbert spaces of analytic functions on the unit disk, we characterize when a function has optimal polynomial approximants given by truncations of a single power series or, equivalently, when the approximants stabilize. We also introduce a generalized notion of optimal approximant and use this to explicitly compute orthogonal projections of 1 onto certain shift invariant subspaces.