## Wavelet theory demystified (2003)

Venue: | IEEE Trans. Signal Process |

Citations: | 45 - 22 self |

### BibTeX

@ARTICLE{Unser03wavelettheory,

author = {Michael Unser and Thierry Blu},

title = {Wavelet theory demystified},

journal = {IEEE Trans. Signal Process},

year = {2003},

pages = {470--483}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract—In this paper, we revisit wavelet theory starting from 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. Index Terms—Approximation order, Besov spaces, Hölder smoothness, multiscale differentiation, splines, vanishing moments, wavelets. I.