A Haer [[Wavelet]] is a [[Continuous Function|non-continuous]] family of [[Wavelet|wavelets]] denoted by $\psi_{a,b}$. $\huge \begin{align} \, \psi_{0,0}(x) &= \begin{cases} 1 & x\leq \frac{1}{2} \\ 0 & x > \frac{1}{2} \end{cases} \\ \psi_{1,0}(x) &= \begin{cases} 1 & x\leq \frac{1}{4} \\ -1 & \frac{1}{4}x \leq \frac{1}{2} \\ 0 \end{cases} \\ \psi_{1,0}(x) &= \begin{cases} 1 & x\leq \frac{1}{4} \\ -1 & \frac{1}{4}x \leq \frac{1}{2} \\ 0 \end{cases} \\ &\vdots \end{align} $ The general recursive definition is the following: $\huge \begin{align} \psi_{a,b}(x) &= \psi_{0,0}\left( 2^{a}x-b \right) \\ \end{align} $ $ \huge \inprod{\psi_{a,b}, \psi_{a,b}} = 2^{-a} $ Where $\inprod{\cdot,\cdot}$ denotes the [[Functional Inner Product]].