Greedy wavelet projections are bounded on BV

Abstract

Abstract. Let BV = BV(R d ) be the space of functions of bounded variation on R d with d ≥ 2. Let ψ λ , λ ∈ ∆, be a wavelet system of compactly supported functions normalized in BV, i.e.,is the set of N indicies λ ∈ ∆ for which |c λ (f )| are largest (with ties handled in an arbitrary way), thenwith C a constant independent of f . This answers in the affirmative a conjecture of Meyer (2001).

Greedy approximation for biorthogonal systems in quasi-Banach spaces

“…We emphasize that (9.8) is considered by some authors as a condition which ensures in a certain sense the optimality of the compression algorithms with respect to the basis (see [40]). We refer the reader to [9,20,33] for the uses of this type of embeddings in the study of non-linear approximation in Banach spaces with respect to bases.…”

Quasi-greedy bases in ℓ (0 < p < 1) are democratic

“…The following lemma summarizes the connections between greedylike bases and embeddings involving sequence Lorentz spaces. We refer the reader to [1,4,7,8,11,37] for the uses of this type of embeddings in the theory of non-linear approximation in Banach spaces with respect to bases. Lemma 2.2 (see [2, Theorem 8.12 and Corollary 8.13]).…”

Greedy approximation for biorthogonal systems in quasi-Banach spaces

“…We emphasize that (8.8) is considered by some authors as a condition which ensures in a certain sense the optimality of the compression algorithms with respect to the basis (see [39]). We refer the reader to [9,20,32] for the uses of this type of embeddings within the study of non-linear approximation in Banach spaces with respect to bases.…”