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A dyadic interval is one cell in a dyadic partition. Its children are the left and right half intervals. Disjoint dyadic intervals at a fixed level produce orthogonal indicator functions and independent Brownian motion increments.

The Haar projection at level $J$ is the orthogonal projection onto the functions constant on level-$J$ dyadic intervals. It replaces a function by its mean on each cell. It can be expressed using level-$J$ Haar scaling functions, or using the constant function and Haar wavelets at levels below $J$.

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 $2^J/n$.