Let Hs be the Sobolev space of exponent s >= 0. Let Vi, i (epsilon) Z,...Vi is contained in Vi-1 is contained in ..., be the ladder spaces of a multiresolution analysis (MRA) in L2(R) associated with a compactly supported scaling function (phi) (epsilon) H(sigma ) which verifies supp(phi) equals [ 0, L-1 ], L (epsilon) N, (sigma) > 1/2. Let f (epsilon) L2(R) be compactly supported with countable many discontinuity points and verifying suppf equals [ t1, t2 ], where 0 < (mu) (suppf) < (infinity) and, for i equals 1,2 and some integer m$_0), ti equals ni(m0)2mo with ni(m0) (epsilon) Z. The goal is to approximate f by a function in Vm which approaches f in some sense as the scale becomes finer, reproduces the values of f at certain equally spaced, arbitrarily close points and is not necessarily the orthogonal projection of f on Vm. Such non-orthogonal projections can be used to start the pyramid algorithm or to approximate f, as an alternative which is better than using the samples themselves in place of the true inner products and faster than integrating for the inner products. The almost uniform convergence feature allows interactive control on the selection of an initial sampling rate that is capable of preserving details down to a desired resolution.