  We study strategies for increasing the precision in the blurring models by
maintaining a complexity in the related numerical linear algebra procedures
(matrix-vector product, linear system solution, computation of eigenvalues
etc.) of the same order of the celebrated Fast Fourier Transform. The key idea
is the choice of a suitable functional basis for representing signals and
images. Starting from an analysis of the spectral decomposition of blurring
matrices associated to the antireflective boundary conditions introduced in [S.
Serra Capizzano, SIAM J. Sci. Comput. 25-3 pp. 1307--1325], we extend the model
for preserving polynomials of higher degree and fast computations also in the
nonsymmetric case.
  We apply the proposed model to Tikhonov regularization with smoothing norms
and the generalized cross validation for choosing the regularization parameter.
A selection of numerical experiments shows the effectiveness of the proposed
techniques.
