Vanishing moments and the approximation power of wavelet expansions

The order of a wavelet transform is typically given by the number of vanishing moments of the analysis wavelet. The Strang-Fix (1971) conditions imply that the error for an orthogonal wavelet approximation at scale a=2^-i globally decays as a^L, where L is the order of the transform. This is why, for a given number of scales, higher order wavelet transforms usually result in better signal approximations. We show that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone. We also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero. These results strengthen Sweldens (see Numer. Math., vol.68, no.3, p.377-401, 1994) earlier analysis and confirm that the approximation power of biorthogonal and (semi-)orthogonal wavelet expansions is essentially the same. Finally, we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines. We also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error