On the Spectral Properties of Matrices Associated with Trend Filters

This note is concerned with the spectral properties of matrices associated with linear smoothers. We derive analytical results on the eigenvalues and eigenvectors of smoothing matrices by interpreting the latter as perturbations of matrices belonging to algebras with known spectral properties, such as the Circulant and the generalised Tau. These results are used to characterise the properties of a smoother in terms of an approximate eigen-decomposition of the associated smoothing matrix.