This paper presents the theoretical underpinnings for a preceding companion work in which new improved accuracy quantifications for noise induced estimation errors were presented. In particular, via the ideas of reproducing kernels and orthonormal parameterisations of the subspaces they represent, this paper develops new methods for evaluating certain quadratic forms in inverse Toeplitz matrices that are instrumental to the quantification of variance error. Additionally, new results on the convergence rates of generalised Fourier expansions are derived and then employed to derive necessary and sufficient conditions for the accuracy of the quantifications of this paper, the preceding companion paper, and the work by other authors.