Definition (Haar function)

Define $$\Psi_{0,0}(x)=\begin{cases} 1 & \text{if }0\le x\le 1\\0 & \text{ else}\end{cases}$$and for $n\in \mathbb{Z}_{+},k\in \{ 0,1,2,\dots,2^{n}-1 \}$ $$\Phi_{n,k}(x)=\begin{cases} 2^{\frac{n}{2}} & \text{if }k2^{-n}\le x< \frac{k+1}{2}2^{-n} \\ -2^{\frac{n}{2}} & \text{if } \frac{k+1}{2}2^{-n}\le x<(k+1)2^{-n} \\ 0 & \text{else} \end{cases}$$