General Haar systems and greedy approximation (Q2717571)

The author generalizes the classical dyadic Haar system by replacing the sequence of dyadic points with an arbitrary dense sequence of points in [0,1]. It is proved that for each general Haar system \(H_T\) obtained in this way there exists a permutation of a subsequence of the dyadic Haar system which is equivalent to \(H_T\) in each \(L^p([0,1])\) \((1<p<\infty)\). Furthermore, it is shown that each normalized general Haar system is a greedy base in \(L^p([0,1])\). In addition, a general Haar system is constructed whose tensor products are greedy bases in each \(L^p([0,1]^d)\) \((1<p<\infty\), \(d\in\mathbb{N})\).