Greedy wavelet projections are bounded on BV

Abstract:

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., $|\psi _\lambda |_{\mathrm {BV}(\mathbb {R}^d)}=1$, $\lambda \in \Delta$. 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 $|c_\lambda (f)|$ 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 $|\mathcal {G}_N(f)|_{\mathrm {BV}(\mathbb {R}^d)}\le C|f|_{\mathrm {BV}(\mathbb {R}^d)}$ with $C$ a constant independent of $f$. This answers in the affirmative a conjecture of Meyer (2001).