Approximate (continuous) functions by step functions - Convergence Rate

Question

I want to approximate a measurable and bounded function $S:[0,1]^2 \rightarrow \mathbb{R}$ by step functions. So, assume we have a uniform partition $$ \{I_j\}_{j=1}^n, \textit{ } j=1,...,n $$ of $[0,1]$. Then a step function $S_n$ with $n$ steps is defined as $$ S_n(u,v) = \sum_{i, j \leq n} a_{i,j}\chi_{I_i}(u) \chi_{I_j}(v). $$ What (continuity-)conditions do we have to assume for $S$ and how are we supposed to choose the coefficients $a_{i,j}$ to achieve "good" convergence rates for the approximation, for instance $$ \|S_n - S\|_{L^{2}} \in \mathcal{O}(f(1/n)) $$ for some $f$ (ideally the identity or better) for $n \rightarrow \infty$.

Answer 1

Since $$ ||S - S_n||_2^2 = \sum_{i,j} \int_{I_i \times I_j} |f - a_{ij}|^2 $$ it's fairly easy to see that the best choice is $$ a_{ij} = \frac{1}{N^2} \int_{I_i \times I_j} f $$ you can prove this by differentiating $$ \alpha \in \mathbb{R} : \varphi(\alpha) = \int_{I_i \times I_j} |f - \alpha|^2$$ and noticing it is convex.