= Cell-average projection
{title2=$U_m=h^{-1}\int_{x_m-h/2}^{x_m+h/2}u(x)\,dx$}
The cell-average projection replaces a function by its average on each grid cell. It is independent of the representative of an <L2 function>, unlike point sampling. The <Cauchy-Schwarz inequality> gives $h|U_m|^2\le\int_{\text{cell }m}|u|^2dx$, hence $h\sum_m|U_m|^2\le\|u\|_{L^2}^2$. Viewed as a piecewise constant function, the projection is the <orthogonal projection> onto the grid's piecewise constant <linear subspace>. Density of continuous compactly supported functions and the contraction estimate give convergence in the <L2 norm> as the mesh tends to zero.
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