The aim of this paper is to provide an analysis of non-linear approximation in the $L_p$-norm $p = d / (d - 1)$ of functions of bounded variation on $\mathbb{R}^d$ with $d > 1$ by polynomials in the Haar system. The exponent $p$ is the natural exponent as it is the correct exponent in the Sobolev inequality. The approximation schemes that we discuss in this paper are mostly related to Haar thresholding and $m$-term approximation. These problems for $d = 2$ are studied in detail in a paper by Cohen, DeVore, Petrushev and Xu. The main aim of this paper is to extend their results to the case $d \geq 2$.

We obtain the optimal order of the $m$-term Haar approximation and prove the stability of Haar thresholding in the BV-norm. As one of the main tools, we establish the boundedness of certain averaging projections in BV.