Wavelet projection ($P_{J}m={\sum }_k⟨m,{\varphi }_{J,k}⟩{\varphi }_{J,k}$)

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 $\Vert m-P_{J}m{\Vert }_2^2={\sum }_{j\ge J,k}|b_{j,k}{|}^2$, which tends to zero by the Parseval identity for a Hilbertian basis. The projection can also be represented by $A_J(x,y)={\sum }_k{\varphi }_{J,k}(x){\varphi }_{J,k}(y)$ as an integral kernel in the Hilbert-space sense.