A Criterion for Convergence of Weak Greedy Algorithms

Abstract

We find a class V of sequences such that the condition τ∉V is necessary and sufficient for convergence of weak greedy algorithm with weakness sequence τ for each f and all Hilbert spaces H and dictionaries D. We denote by V the class of sequences x={x _{k k=1} ^∞, x _k ≥0, k=1,2,..., with the following property: there exists a sequence 0=q ₀<q ₁<⋅⋅⋅ such that ∑ _s=1 ^∞2^s/Δq _s )<∞ and ∑ _s=1 ^∞2^−s∑ _k=1 ^q s ^x _k ²<∞, where Δq _s :=q _s −q _s−1.