Algorithms and Error Bounds for Multivariate Piecewise Constant Approximation

Let Ω be a bounded domain in R, d ≥ 2. Suppose that ∆ is a partition of Ω into a finite number of subsets ω ⊂ Ω called cells, where the default assumptions are just these: |ω| := meas(ω) > 0 for all ω ∈ ∆, |ω ∩ ω′| = 0 if ω 6= ω, and ∑ ω∈∆ |ω| = |Ω|. For a finite set D we denote its cardinality by |D|, so that |∆| stands for the number of cells ω in ∆. Given a function f : Ω → R, we are interested in the error bounds for its approximation by piecewise constants in the space