Greedy Wavelet Projections are Bounded on BV

"Greedy Wavelet Projections are Bounded on BV"

Let BV = BV(IR^d) be the space of functions of bounded variation on IRd with d ≥ 2. Let Ψ_λ, λ ∈ Δ, be a wavelet system of compactly supported functions normalized in BV, i.e. |Ψ_λ|BV(IR^d) = 1, λ ∈ Δ. Each f ∈ BV has a unique wavelet expansion ∑_λ∈Δ c_λ(f)Ψ_λ with convergence in L₁(IR^d). If ^N(f) is the set of N indicies λ ∈ Δ for which |c_λ(f)| are largest (with ties handled in an arbitrary way), then G_N(f) := ∑_λ∈^N(f) c_λ(f)Ψ_λ is called a greedy approximation to f. It is shown that |G_N(f)|_BV(IR^d) ≤ C|f|_BV(IR^d) with C a constant independent of f. This answers in the affirmative a conjecture of Meyer [15] (see p. 49).