= Haar refinement identity
{c}
{title2=$V_{j+1}=V_j\oplus W_j$}
The normalized <Haar scaling functions> and wavelets obey
$$
\varphi_{j,k}=2^{-1/2}(\varphi_{j+1,2k}+\varphi_{j+1,2k+1}),\qquad
\psi_{j,k}=2^{-1/2}(\varphi_{j+1,2k}-\varphi_{j+1,2k+1}).
$$
This orthogonal two-by-two transformation gives the <multiresolution analysis> decomposition $V_{j+1}=V_j\oplus W_j$. Iterating it expresses a <Haar approximation> either through level-$j$ cell averages or through coarse averages and all wavelet details at lower levels. The identities remain valid locally for coefficient integrals of <locally integrable functions>, without a global $L^2$ assumption.
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