For a real function $f$ on $\mathbb{R}$, define $e_n(f)$ to be the infimum of the $L_1$ distance between $f$ and piecewise constant functions on the subdivision of $\mathbb{R}$ into intervals of length $1/n$ ($0$ being one of the intervals bound). For $c>0$, does the space of $f$ such that $e_n(f) = O(n^{-c})$ have a name? What's a good reference for related notions?
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I think this is answered
DeVore, Ronald A. "Nonlinear approximation." Acta numerica 7 (1998): 51-150
in Section 3.1: The space of functions for which $e_n(f) = O(n^{-c})$ is the space $\mathrm{Lip}(c,L^1(0,1))$ and this space consists of the function $f$ for which $$ \|f(\cdot+h)-f\|_{L^1(0,1-h)} \leq Mh^c $$ for some constant $M$. I do not know how these spaces are called, though.
