Greedy wavelet projections are bounded on BV

Abstract

Abstract. Let BV = BV(Rd) be the space of functions of bounded variation on Rd with d ≥ 2. Let ψλ, λ ∈ ∆, be a wavelet system of compactly supported functions normalized in BV, i.e., |ψλ | BV(Rd) =1,λ ∈ ∆. Each f ∈ BV has a unique wavelet expansion ∑ λ∈ ∆ cλ(f)ψλ with convergence in L1(Rd). If ΛN (f) isthesetofN indicies λ ∈ ∆forwhich|cλ(f) | are largest (with ties handled in an arbitrary way), then GN (f): = ∑ λ∈ΛN (f) cλ(f)ψλ is called a greedy approximation to f. Itisshownthat|GN (f) | BV(Rd) ≤ C|f | BV(Rd) with C a constant independent of f. This answers in the affirmative a conjecture of Meyer (2001). 1.