Haar system

Remarks

The function $h_{n,k}$ is supported on the dyadic interval $[k 2^{-n}, (k+1) 2^{-n}]$, where it equals $2^{n/2}$ on the left half and $-2^{n/2}$ on the right half. The Haar system is an orthonormal basis of $L^2([0, 1])$ (see Lp space). On the whole line, the functions $2^{j/2}\psi(2^j t - k)$ with $j, k \in \Z$ form an orthonormal basis of $L^2(\R)$.

Example: $h_{1,0}$ and $h_{1,1}$ are orthogonal, since their supports $[0, \tfrac12]$ and $[\tfrac12, 1]$ meet in one point. Also $h_{0,0}$ and $h_{1,0}$ are orthogonal: on $[0, \tfrac12]$ the function $h_{0,0}$ is constant $1$, and $h_{1,0}$ has integral $0$ there. And $\norm{h_{1,0}}^2 = 2 \cdot \tfrac12 = 1$.

Non-example: the family of all $h_{n,k}$ without the constant function $1$ is orthonormal but not a basis of $L^2([0, 1])$. Indeed, the constant $1$ is orthogonal to every $h_{n,k}$, since each $h_{n,k}$ has integral $0$.