Haar System and Wavelets

• supported in $\text{[}a,b]$ means that the function is integrable in the interval and zero outside of that interval.
• A refinement equation expresses a function as a linear combination of scaled and shifted copies of itself, typically used to define wavelets or scaling functions at multiple resolutions.

Haar Functions

The Haar Scaling Function (Box Function)

\varphi(x) = \left{\begin{matrix} 1, && 0 \le x < 1 \\ 0, && x < 0 \ \text{or}\ x \ge 1 \\ \end{matrix}\right.

The Haar Wavelet Function

\psi(x)=\left{\begin{matrix} 1, && 0 \le x < \frac{1}{2} \\ -1, && \frac{1}{2} \le x < 1 \\ 0, && x < \ \text{or} \ x \ge 1 \end{matrix}\right.

The Haar Scaling function for arbitrary integers on $\text{[}k,k+1)$

\varphi\_{k}(x) = \left{\begin{matrix} 1, && k \le x < k+1 \\ 0, && x < k \ \text{or}\ x \ge k + 1 \\ \end{matrix}\right.

Haar Wavelet Function for arbitrary integers

\psi\_{n, k} = 2^{\frac{2}{n}}\psi(2^nx-k)=\left{\begin{matrix} 2^{\frac{2}{n}} && 2^{-n}k\le x <2^{-n}\left( k+\frac{1}{2} \right) \\ -2^{\frac{2}{n}} && 2^{-n}\left( k+\frac{1}{2}\right) \le x < 2^{-n} (k+1) \\ 0 && x < 2^{-n}k \ \text{or} \ \ge 2^{-n}(k+1) \end{matrix}\right.

$n$ is scale. $k$ is position.

Haar Coefficients

$$\begin{array}{c}a\_k=⟨f,\phi \_k⟩=\int \_k^{k+1}f(x),dx\\ b\_n,k=2^{n/2}\int \_{-\infty }^{\infty }f(x),\psi (2^{n}x-k),dx=2^{n/2}\left(\int \_{2^{-n}k}^{2^{-n}(k+1/2)}f(x),dx-\int \_{2^{-n}(k+1/2)}^{2^{-n}(k+1)}f(x),dx\right)\end{array}$$

Approximations

Finite Linear Combination of Box Functions

$$p\_0(x)=\sum \_{k=-\infty }^{\infty }a\_k\phi \_k(x)=\sum \_{k=M}^{N-1}a\_k\phi \_k(x)$$

If we take $M<N$ as integers and $f$ is zero outside of $\text{[}M,N]$ then $a\_k=0$ for all $k<M$ and $k\ge N$. In such case the version with $M$ and $N$ can be used.

Finite Linear Combinations of Wavelets (Detail)

The function $q\_0$ represents the details in $f$ at resolution level zero.

$$q\_0(x)=\sum \_{k=-\infty }^{\infty }b\_0,k\psi \_0,k(x)$$

Increasing the Detail

The approximation is made better by adding $q\_0$ to $p\_0$ to obtain a better approximation $p\_0$. $p\_1$ approximates $f$ with resolution level one.

$$p\_1(x)=p\_0(x)+q\_0(x)$$

More Details

Increasing the detail further requires to add more detail functions $q\_n(x)$ of resolution $n$ to the approximation $p\_n(x)$. Such that the next resolution is given by $p\_n+1(x)=p\_n(x)+q\_n(x)$.

The Resolution Function at Resolution $n$

$$q\_n(x)=\sum \_{k=-\infty }^{\infty }b\_n,k\psi \_n,k(x)$$