Multiresolution

Definition. $L^2(\mathbb{R})$

$$L^2(\mathbb{R})=\{f:\mathbb{R}\mapsto \mathbb{C}|{\int }_{\mathbb{R}}|f(x){|}^{2}dx<\infty \}$$

Definition. Multiresolution Analysis

A multiresolution analysis for $L^2(\mathbb{R})$ consists of

• a sequence of closed subspaces $(V_j{)}_{j\in \mathbb{Z}}$ of $L^2(\mathbb{R})$, and
• a scaling function $\varphi \in V_0$ ,

such that the following conditions holds,

• $\{\varphi (\cdot -k){\}}_{k\in \mathbb{Z}}$ constitute an orthonomal basis of $V_0$.
• $\forall i\in \mathbb{Z},V_{j-1}\subset V_j$,
• $\overline{\cup V_j}=L^2(\mathbb{R})$,
• $\cap V_j=\{0\}$,
• $f\in V_0⟺f(2^j\cdot )\in V_j$,

The scaling function $\varphi$ in multiresolution analysis determines the spaces $V_j$ uniquely.

Definition. Wavelet space

For any $j\in \mathbb{Z}$, let $W_j$ denote the orthogonal complement of $V_j$ with respect to $V_{j+1}$, i.e.

$$W_j=\{f\in V_{j+1}|⟨f,g⟩=0,\forall g\in V_j\}.$$

For each $j\in \mathbb{Z}$, the closed subspaces,

$$V_j=V_{j-1}\oplus W_{j-1}=\cdots \oplus W_{j-2}\oplus W_{j-1}$$

Any wavelet generate a direct sum decomposition of $L^2(\mathbb{R})$.

Basis for $V_j$, $\{{\varphi }_k^j=2^{j/2}\varphi (2^{j}x-k){\}}_{k\in \mathbb{Z}}$

$\varphi (x)={\sum }_{k\in \mathbb{Z}}{p}_k\varphi (2x-k)$

${\varphi }_l^{j-1}=2^{-1/2}{\sum }_{k\in \mathbb{Z}}{p}_{k-2l}{\varphi }_k^j$

Basis for $W_j$, $\{{\psi }_k^j=2^{j/2}\psi (2^{j}x-k){\}}_{k\in \mathbb{Z}}$

$\psi (x)={\sum }_{k\in \mathbb{Z}}(-1{)}^k\overline{p_{l-k}}\psi (2x-k)$

${\psi }_l^{j-1}=2^{-1/2}{\sum }_{k\in \mathbb{Z}}(-1{)}^k\overline{p_{1-(k-2l)}}{\psi }_k^j$

Decomposition formulas

Denote $f_k^j=⟨f,{\varphi }_k^j⟩$ and $g_k^j=⟨f,{\psi }_k^j⟩$,

$$\{\begin{array}{l}{f}_l^{j-1}=2^{-1/2}{\sum }_k\overline{p_{k-2l}}{f}_k^j,\\ g_l^{j-1}=2^{-1/2}{\sum }_k(-1{)}^k{p}_{1-(k-2l)}{f}_k^j,\end{array}$$

Reconstruction formulas

$$f_k^j=\sum _l{p}_{k-2l}{f}_l^{j-1}+\sum _l(-1{)}^k\overline{p_{1-(k-2l)}}{g}_l^{j-1}$$

Proposition.

The function $\varphi \in L^2(\mathbb{R})$ generate a multiresolution analysis. Let $\psi \in L^2(\mathbb{R})$ and $\{\psi (\cdot -k){\}}_{k\in \mathbb{Z}}$ is an orthonormal basis for $W_0$, then

• the functions $\{\psi (2^j\cdot -k){\}}_{k\in \mathbb{Z}}$ form an orthonormal basis for $W_j$,
• $\psi$ is wavelet, i.e., the functions $\{\psi (2^j\cdot -k){\}}_{j,k\in \mathbb{Z}}$ form an orthonormal basis for $W_j$,
• the functions $\{\varphi (\cdot -k){\}}_{k\in \mathbb{Z}}\cup \{\psi (2^j\cdot -k){\}}_{j\in \mathbb{N},k\in \mathbb{Z}}$ form an orthonormal basis for $L^2(\mathbb{R})$.

Proposition.

The function $\varphi \in L^2(\mathbb{R})$ generate a multiresolution analysis. Then there exist a 1-periodic function $H\in L^2(0,1)$ such that

$$\hat{\varphi }(2t)=H(t)\hat{\varphi }(t),t\in \mathbb{R}.$$

where

$$H(t)=\sum _{k\in \mathbb{R}}{c}_k{e}^{2\pi ikt}$$

Let

$$G(t)=\overline{H(t+\frac{1}{2})}{e}^{-2\pi it}$$

define the function $\psi$

$$\hat{\psi }(2t)=G(t)\hat{\varphi }(t)$$

then

• the functions $\{\psi (2^j\cdot -k){\}}_{k\in \mathbb{Z}}$ form an orthonormal basis for $W_j$,
• $\psi$ is wavelet, i.e., the functions $\{\psi (2^j\cdot -k){\}}_{j,k\in \mathbb{Z}}$ form an orthonormal basis for $W_j$,

$$G(t)=\overline{H(t+\frac{1}{2})}{e}^{-2\pi it}=\sum _{k\in \mathbb{R}}{\bar{c}}_k{e}^{-2\pi ik(t+\frac{1}{2})-2\pi it}$$

Proposition

If

$$G(t)=\sum _{k\in \mathbb{R}}{d}_k{e}^{2\pi ikt}$$

then

$$\psi (x)=2\sum _{k\in \mathbb{Z}}{d}_k\varphi (2x+k),x\in \mathbb{R}.$$

Compactly supported wavelet means that only few coeficients of $d_k$ are non-zeros.

Theorem

Let $\varphi \in L^2(\mathbb{R})$, $\{\varphi (\cdot -k){\}}_{k\in \mathbb{Z}}$ is an orthonormal system iff.

$$\sum _{k\in \mathbb{Z}}|\hat{\varphi }(t+k){|}^2=1,t\in \mathbb{R}.$$