The aim of this paper is to provide an analysis of non-linear approximation in the \(L_p\) norm, \(p=d/(d-1)\), on the class \(BV({\mathbb R}^d)\) of functions of bounded variation. The approximation schemes discussed in the paper use Haar functions. The first scheme is the \(m\)-term approximation in which, for a given \(f \in BV({\mathbb R}^d)\) and a natural number \(m\), one selects \(m\) Haar functions \(h_k\) and the coefficients \(a_k\) to obtain a good approximation \(\sum_1^m a_k h_k\) to \(f\). The second scheme provides an approximation to \(f\) by retaining only those terms in the Haar basis expansion of \(f\) whose coefficients exceed some fixed threshold \(\epsilon>0\). For \(d=2\) these problems were studied in [A. Cohen, R. DeVore, P. Petrushev and H. Xu, Am. J. Math. 121, 587–628 (1999; Zbl 0931.41019)].
The main theorem of the paper gives natural sufficient conditions on the sequence of sets \((K_j)\) to ensure that \[ Pf(x)=\begin{cases} {1 \over | K_j| }\int_{K_j}f(t)\,dt &\text{ if }x \in K_j\\ f(x)&\text{ if }x \notin \cup K_j\end{cases} \] is a bounded projection in \(BV({\mathbb R}^d)\).