Greedy wavelet projections are bounded on BV

Let $\mathrm{BV}=\mathrm{BV}(\mathbb{R}^d)$ be the space of functions of bounded variation on $\mathbb{R}^d$ with $d\ge 2$. Let $\psi_\lambda$, $\lambda\in\Delta$, be a wavelet system of compactly supported functions normalized in $\mathrm{BV}$, i.e., $\vert\psi_\lambda\vert _{\mathrm{BV}(\mathbb{R}^d)}=1$, $\lambda\in\Delta$. Each $f\in \mathrm{BV}$ has a unique wavelet expansion $\sum_{\lambda\in\Delta} c_\lambda(f)\psi_\lambda$ with convergence in $L_1(\mathbb{R}^d)$. If $\Lambda_N(f)$ is the set of $N$ indicies $\lambda\in\Delta$ for which $\vert c_\lambda(f)\vert$ are largest (with ties handled in an arbitrary way), then $\mathcal{G}_N(f):=\sum_{\lambda\in\Lambda_N(f)}c_\lambda(f)\psi_\lambda$ is called a greedy approximation to $f$. It is shown that $\vert\mathcal{G}_N(f)\vert _{\mathrm{BV}(\mathbb{R}^d)}\le C\vert f\vert _{\mathrm{BV}(\mathbb{R}^d)}$ with $C$ a constant independent of $f$. This answers in the affirmative a conjecture of Meyer (2001).