Wavelet theory demystified - Signal Processing, IEEE Transactions on
Michael Unser, Thierry Bluthe 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.