Greedy wavelet projections are bounded on BV

Abstract

Let B V = B V ( R d ) \mathrm {BV}=\mathrm {BV}(\mathbb {R}^d) be the space of functions of bounded variation on R d \mathbb {R}^d with d ≥ 2 d\ge 2 . Let ψ λ \psi _\lambda , λ ∈ Δ \lambda \in \Delta , be a wavelet system of compactly supported functions normalized in B V \mathrm {BV} , i.e., | ψ λ | B V ( R d ) = 1 |\psi _\lambda |_{\mathrm {BV}(\mathbb {R}^d)}=1 , λ ∈ Δ \lambda \in \Delta . Each f ∈ B V f\in \mathrm {BV} has a unique wavelet expansion ∑ λ ∈ Δ c λ ( f ) ψ λ \sum _{\lambda \in \Delta } c_\lambda (f)\psi _\lambda with convergence in L 1 ( R d ) L_1(\mathbb {R}^d) . If Λ N ( f ) \Lambda _N(f) is the set of N N indicies λ ∈ Δ \lambda \in \Delta for which | c λ ( f ) | |c_\lambda (f)| are largest (with ties handled in an arbitrary way), then G N ( f ) := ∑ λ ∈ Λ N ( f ) c λ ( f ) ψ λ \mathcal {G}_N(f):=\sum _{\lambda \in \Lambda _N(f)}c_\lambda (f)\psi _\lambda is called a greedy approximation to f f . It is shown that | G N ( f ) | B V ( R d ) ≤ C | f | B V ( R d ) |\mathcal {G}_N(f)|_{\mathrm {BV}(\mathbb {R}^d)}\le C|f|_{\mathrm {BV}(\mathbb {R}^d)} with C C a constant independent of f f . This answers in the affirmative a conjecture of Meyer (2001).