Multiple-scale decomposition for the Zlamal approximation

For the Zlamal approximation (a piecewise-polynomial continuous approximation of degree at most two), it is proved that the space of approximating functions obtained by subdividing a triangulation contains the space corresponding to the given triangulation. Formulas for the multiple-scale decomposition (that is, the decomposition of old basis functions in the new ones) are explicitly written in the case of placing one additional vertex on one edge of the initial triangulation. The cases of placing a vertex on a boundary or an interior edge are considered. The obtained formulas can also be used when several vertices are added to sufficiently distant triangles, because the operation under consideration and its influence on the coefficients in the decomposition of the approximating function in the standard Zlamal basis are local. The local bases of the additional terms W in the decomposition of the new space of approximating functions into the direct sum of the old space and W are specified (in particular, in the cases of placing a new vertex on a boundary or an interior edge). For these bases, decomposition formulas and reconstructions of the wavelet transform are explicitly written. All of the formulas were tested by using the MuPAD 2.5.3 computer algebra system running under Linux.