  Linear approximation approaches suffer from Gibbs oscillations when
approximating functions with singularities. ENO-SR resolution is a local
approach avoiding oscillations and with a full order of accuracy, but a loss of
regularity of the approximant appears. The goal of this paper is to introduce a
new approach having both properties of full accuracy and regularity. In order
to obtain it, we propose a three-stage algorithm: first, the data is smoothed
by subtracting an appropriate non-smooth data sequence; then a chosen high
order linear approximation operator is applied to the smoothed data and
finally, an approximation with the proper singularity structure is reinstated
by correcting the smooth approximation with the non-smooth element used in the
first stage. We apply this approach to both cases of point-value data and of
cell-average data, using the 4-point subdivision algorithm in the second stage.
Using the proposed approach we are able to construct approximations with high
precision, with high piecewise regularity, and without diffusion nor
oscillations in the presence of discontinuities.
