= Wavelet projection
{title2=$P_Jm=\sum_k\langle m,\phi_{J,k}\rangle\phi_{J,k}$}
The wavelet projection is the <orthogonal projection> onto the approximation space $V_J$. Iterating $V_{j+1}=V_j\oplus W_j$ expresses it as the coarse part of the <wavelet series> plus details at levels below $J$. <Orthogonality> gives $\|m-P_Jm\|_2^2=\sum_{j\geq J,k}|b_{j,k}|^2$, which tends to zero by the <Parseval identity for a Hilbertian basis>. The projection can also be represented by $A_J(x,y)=\sum_k\phi_{J,k}(x)\phi_{J,k}(y)$ as an <integral kernel> in the Hilbert-space sense.
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