Compactly supported wavelets

Ingrid Daubechies defined the class of compactly supported wavelets [15]. Briefly, let $\varphi$ be a solution of the scaling relation

$$\varphi(x)=\sum_{k=0}^N a_k\varphi(2x-k).$$

The $a_k$ are a collection of coefficients that categorize the specific wavelet basis. The expression $\varphi$ is called the scaling function.

The normalization $\int \varphi dx = 1$ of the scaling function implies the condition

$$\sum a_k = 2.$$

The translates of $\varphi$ are required to be orthonormal

$$\int \varphi(x-k)\varphi(x-m)=\delta_{k,m}.$$

From the scaling relation this implies the condition

$$\sum_{k=0}^N a_k a_{k-2m}=\delta_{0m}.$$

For coefficients verifying the above two conditions, the functions consisting of translates and dilations of the scaling function, $\varphi(2^jx-k)$, form a complete, orthogonal basis for square integrable functions on the real line, $L^2(R)$.

If only a finite number of the $a_k$ are nonzero then $\varphi$ will have compact support.

Smooth scaling functions arise as a consequence of the degree of approximation of the translates. The conditions that the polynomials $1,x,\cdots,x^{p-1}$ be expressed as linear combinations of the translates of $\varphi(x-k)$ is implied by the condition $\sum(-1)^k k^m a_k = 0$ for $m=0,1,\cdots,p-1.$

The compactly supported wavelet is defined by the equation

$$\psi(x) = \sum (-1)^k a_{1-k}\phi(2x-k)$$

The translates of the scaling function and wavelet define orthogonal subspaces. i.e.

$$\int\phi(x)\psi(x-m)dx=\sum(-1)^k a_{1-k} a_{k-2m}=0.$$

The orthogonal subspaces:

$$V_j=\{2^{j/2}\phi(2^jx-m);m=\cdots,-1,0,1,\cdots\}$$

$$W_j=\{2^{j/2}\psi(2^jx-m);m=\cdots,-1,0,1,\cdots\}$$

are related by the condition

$$V_{j+1} = V_j \oplus W_j.$$

This is the basis of Mallat, or Wavelet, transform

$$V_{j+1} = V_0 \oplus W_0 \oplus W_1 \oplus \cdots \oplus W_j$$

where $V_0 \subset V_1 \subset \cdots \subset V_{j+1}$.

In Figure 1 we see pictured an example of a compactly supported scaling function and its associated fundamental wavelet function.

Figure 1: Daubechies' Scaling and Wavelet Functions for $N=6$ with support on $[0,5]$

By rescaling and translations we obtain a complete orthonormal system for $L^2({\bf R})$ which has sufficient smoothness to also be a basis for $H^1({\bf R})$. This wavelet system then yields a basis for solution methods for second order elliptic boundary problems on intervals on the real line. The illustrated example, which is due to Daubechies [15], has fundamental support $[0,5]$.