Greedy algorithm with gaps

We generalize the well-known greedy approximation algorithm, by allowing gaps in the approximating sequence. We give examples of bases which are “quasi-greedy with gaps,” in spite of failing to be quasi-greedy in the usual sense. However, we also show that for some classical bases (such as the normalized Haar basis in L1, and the trigonometric basis in Lp for p≠2), the greedy algorithm may diverge, even if gaps are introduced into the approximating sequence.