Wavelet theory demystified - Signal Processing, IEEE Transactions on

the representation of a scaling function as the convolution of a

B-spline (the regular part of it) and a distribution (the irregular

or residual part). This formulation leads to some new insights on

wavelets and makes it possible to rederive the main results of the

classical theory—including some new extensions for fractional

orders—in a self-contained, accessible fashion. In particular, we

prove that the B-spline component is entirely responsible for five

key wavelet properties: order of approximation, reproduction

of polynomials, vanishing moments, multiscale differentiation

property, and smoothness (regularity) of the basis functions.

We also investigate the interaction of wavelets with differential

operators giving explicit time domain formulas for the fractional

derivatives of the basis functions. This allows us to specify a

corresponding dual wavelet basis and helps us understand why

the wavelet transform provides a stable characterization of the

derivatives of a signal. Additional results include a new peeling

theory of smoothness, leading to the extended notion of wavelet

differentiability in the -sense and a sharper theorem stating

that smoothness implies order.