For a vector $u\in\mathbf{R}^N$ let's denote $\pi_N(u)$ the unique piecwise linear and $1$-periodic function matching the components of $u$ on the discretization $x_k = \frac{k}{N}$ of the unit interval $[0,1]$. Similarly we denote $\sigma_N(u)$ the unique piecewise constant and $1$-periodic function taking value $u_k$ on $[x_{k-1},x_k]$.
Thanks to the equality $\mathbf{1}_{[-1,0]}\star\mathbf{1}_{[0,1]}(x) = (1-|x|)^+$, one can check that $\pi_M(u) = \sigma_M(u)\star\rho_M$, where $\rho_M$ is the approximate identity generated by $\mathbf{1}_{[0,1]}$.
I would like to know if a similar relationship between $\sigma_M(u)$ and $\pi_M(u)$ persist in higher dimension, for instance in dimension $2$, for a uniform triangulation of the unit square $[0,1]^2$.
