# Contents

## Idea

Daubechies wavelets are a family of wavelets that form an orthonormal basis via a multiresolution analysis and are optimal in a certain sense that we explain now: We would like to have a wavelet base such that for most signals $f$ most of the wavelet coefficients are small or zero, so that the signal can be approximately represented and reconstructed with a small set of coefficients. There are two properties of a mother wavelet that help in this regard:

• a minimal, compact support: the smaller the support of the wavelet is, the less of the signal it picks up in a certain wavelet coefficient.

• vanishing moments: when a wavelet is orthogonal to the polynomials $1, t, t^2, ..., t^n$, and we approximate the signal $f$ on the support of the wavelet with its Taylor series, then the wavelet will ignore the first terms of the Taylor series. The better the signal is approximated by its Taylor series, the smaller the wavelet coefficients will be.

For a wavelet these two properties are not independent, for a given number of vanishing moments, there is a minimum, nonzero length $b - a$ of the support $[a, b]$ of the mother wavelet. This is a famous theorem by Ingrid Daubechies. The Daubechies wavelets are optimal in the sense that they have the minimal support for a prescribed number of vanishing moments. There is one Daubechies wavelet for every $p \in \mathbb{N}_+$, where $p$ is the number of vanishing moments.

Caveat: Daubechies wavelets are not smooth and asymmetric. Depending on the application at hand, this can outweigh the theoretical advantages.

## Details

###### Daubechies: Vanishing Moments and Support

If $\psi$ is a wavelet with p vanishing moments that generates an orthonormal basis of $L^2(\mathbb{R})$, then it has a support of size larger than or equal to $2p-1$. A Daubechies wavelet has a minimum-size support equal to $[-p+1, p]$. The support of the corresponding scaling function $\phi$ is $[0, 2p+1]$.