Haar wavelet

Example

The Haar wavelet is the function $\psi \colon \R \to \R$ with $$\psi(x) = \begin{cases} 1 & 0 \le x < \tfrac12, \\ -1 & \tfrac12 \le x < 1, \\ 0 & \text{otherwise.} \end{cases}$$ It is an orthonormal wavelet: the functions $\psi_{j,k}(x) = 2^{j/2} \psi(2^j x - k)$, $j, k \in \Z$, form an orthonormal basis of $L^2(\R)$.

Check of orthonormality. $\psi_{j,k}$ is supported on the dyadic interval $I_{j,k} = [k 2^{-j}, (k+1) 2^{-j})$ and $\norm{\psi_{j,k}}_2^2 = 2^j \cdot 2^{-j} = 1$. Two dyadic intervals are either disjoint or nested. If they are disjoint, the product $\psi_{j,k} \psi_{j',k'}$ is $0$. If $I_{j',k'}$ lies strictly inside $I_{j,k}$, then it lies inside one half of $I_{j,k}$, where $\psi_{j,k}$ is constant, and $\psi_{j',k'}$ has integral $0$.

Completeness comes from the Haar multiresolution analysis, whose space $V_j$ consists of the functions in $L^2(\R)$ that are constant on the intervals $I_{j,k}$. The scaling function is $\varphi = 1_{[0, 1)}$, with $\varphi(x) = \varphi(2x) + \varphi(2x - 1)$ and $\psi(x) = \varphi(2x) - \varphi(2x - 1)$. On $[0, 1]$, the constant $1$ together with the $\psi_{j,k}$ with $j \ge 0$ and $0 \le k < 2^j$ is the Haar system.

The Haar wavelet has exactly one vanishing moment: $\int \psi = 0$, while $\int_\R x \psi(x) \, dx = \tfrac18 - \tfrac38 = -\tfrac14$. It is discontinuous, so for smooth functions its expansions converge slowly compared with smoother wavelets.