ASYMPTOTIC ERROR EXPANSION OF WAVELET APPROXIMATIONS OF SMOOTH FUNCTIONS .2.

We generalize earlier results concerning an asymptotic error expansion of wavelet approximations. The properties of the monowavelets, which are the building blocks for the error expansion, are studied in more d etail, and connections between spline wavelets and Euler and Bernoulli polynomials are pointed out. The expansion is used to compare the err or for different wavelet families. We prove that the leading terms of the expansion only depend on the multiresolution subspaces V(j) and no t on how the complementary subspaces W(j) are chosen. Consequently, fo r a fixed set of subspaces V(j), the leading terms do not depend on th e fact whether the wavelets are orthogonal or not. We also show that D aubechies' orthogonal wavelets need, in general, one level more than s pline wavelets to obtain an approximation with a prescribed accuracy. These results are illustrated with numerical examples.