Haar scaling function (${\varphi }_{J,k}=2^{J/2}{1}_{I_{J,k}}$)

At level $J$, a Haar scaling function is the normalized indicator function $2^{J/2}{1}_{I_{J,k}}$ of a dyadic interval. The functions at that level form an orthonormal basis of the cellwise-constant subspace. Their normalization is what turns white-noise coefficient variances into $1/n$ and cellwise variances into $2^J/n$.