= Haar scaling function
{c}
{title2=$\phi_{J,k}=2^{J/2}\mathbf1_{I_{J,k}}$}
= Haar father wavelet
{c}
{synonym}
At level $J$, a <Haar scaling function> is the normalized <indicator function> $\phi_{J,k}=2^{J/2}\mathbf1_{I_{J,k}}$ of the <dyadic interval> $I_{J,k}=[k2^{-J},(k+1)2^{-J})$. Its <L2 norm> is one because its squared height is $2^J$ and its support length is $2^{-J}$.
For a real <square-integrable function> $f$, its coefficient is
$$
\langle f,\phi_{J,k}\rangle
=2^{J/2}\int_{I_{J,k}}f(x)\,dx
=2^{-J/2}\overline f_{I_{J,k}}.
$$
Multiplying the coefficient by the <Haar scaling function> restores the <cell average> on that interval. The disjoint <supports> make the functions at one level an <orthonormal basis> of the cellwise-constant subspace. This normalization is also what turns coefficient <variances> in the <Gaussian white noise model> into $1/n$ and cellwise variances into $2^J/n$ in the <Haar projection estimator in Gaussian white noise>.
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