Kernel Methods for Regression in Continuous Time over Subsets and Manifolds

The paper derives the same two-term PE-based error bound for the projected/Galerkin regressor on a subspace (HS in the paper) as the candidate solution does for a general closed subspace V. Both use the same steps: (i) projected normal equations, (ii) subtracting the identity for ΠV G to get an error equation, (iii) a PE-induced lower spectral bound yielding a bounded-inverse estimate, and (iv) the operator-norm bound ‖Tφ(0,t)‖ ≤ K̄² ν([0,t]), culminating in the identical inequality when ν is Lebesgue and t = mΔ. The paper additionally contrasts the unprojected estimator in H (which produces a third “drift” term) with the projected one, matching the candidate’s explanation of where the projection is used to remove the drift. Minor differences are only in notation and in the paper’s explicit discussion of when PE on HS forces HS to be finite dimensional.