Lebesgue-Type Inequalities for Quasi-greedy Bases

Abstract

We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N-term approximation ∥x−GNx∥ and the best N-term approximation σN(x) is controlled by max{μ(N),kN}, where μ(N) and kN are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is O(logN). We show with various examples that these bounds are actually attained.