General Haar systems and greedy approximation

Anna Kamont
2001 Studia Mathematica  
We show that each general Haar system is permutatively equivalent in L p ([0, 1]), 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in L p ([0, 1]), 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each L p ([0, 1] d ), 1 < p < ∞, d ∈ N. This is in contrast to [11] , where it has been shown that the tensor products of the dyadic Haar system are not
more » ... dy bases in L p ([0, 1] d ) for 1 < p < ∞, p = 2 and d ≥ 2. We also note that the above-mentioned general Haar system is not permutatively equivalent to the whole dyadic Haar system in any L p ([0, 1]), 1 < p < ∞, p = 2.
doi:10.4064/sm145-2-5 fatcat:iu4ygwxghjdsrcxsmrz4pm7ryy