For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by $\psi _ { j } ( t ) = \max ( 0,1 - j | t | )$. For integers $0 \leqslant k < j$, $\psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right)$. We are given $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$ and $j \geqslant 2$ an integer, and $$S _ { j } ( f ) \left( \theta _ { 1 } , \theta _ { 2 } \right) = \sum _ { k _ { 1 } = 0 } ^ { j - 1 } \sum _ { k _ { 2 } = 0 } ^ { j - 1 } f \left( \frac { k _ { 1 } } { j } , \frac { k _ { 2 } } { j } \right) \psi _ { j , k _ { 1 } } \left( \theta _ { 1 } \right) \psi _ { j , k _ { 2 } } \left( \theta _ { 2 } \right) .$$
Let $j \geqslant 2 , k _ { 1 }$ and $k _ { 2 }$ be two integers such that $0 \leqslant k _ { 1 } , k _ { 2 } < j$, and $$\theta \in \left[ \frac { k _ { 1 } } { j } , \frac { k _ { 1 } + 1 } { j } \left[ \times \left[ \frac { k _ { 2 } } { j } , \frac { k _ { 2 } + 1 } { j } [ . \right. \right. \right.$$
Express $S _ { j } ( f ) ( \theta )$ as a barycenter of the points $f \left( \frac { \ell _ { 1 } } { j } , \frac { \ell _ { 2 } } { j } \right)$ where $\ell _ { 1 } \in \left\{ k _ { 1 } , k _ { 1 } + 1 \right\}$ and $\ell _ { 2 } \in \left\{ k _ { 2 } , k _ { 2 } + 1 \right\}$. Deduce that $\left\| S _ { j } ( f ) - f \right\| _ { \infty } \rightarrow 0$ when $j \rightarrow + \infty$.