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?



Name of a function space
I think this is answered 



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> DeVore, Ronald A. "Nonlinear approximation." Acta numerica 7 (1998): 51-150
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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.


