Asymptotic error expansion of wavelet approximations of smooth functions II

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 detail, and connections between spline wavelets and Euler and Bernoulli polynomials are pointed out. The expansion is used to compare the error for different wavelet families. We prove that the leading terms of the expansion only depend on the multiresolution subspaces Vj and not on how the complementary subspaces Wj are chosen. Consequently, for a fixed set of subspaces Vj , the leading terms do not depend on the fact whether the wavelets are orthogonal or not. We also show that Daubechies’ orthogonal wavelets need, in general, one level more than spline wavelets to obtain an approximation with a prescribed accuracy. These results are illustrated with numerical examples.