Haar refinement identity ($V_{j+1}=V_j\oplus W_j$)

The normalized Haar scaling functions and wavelets obeyThis 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.