General Haar systems and greedy approximation

Anna Kamont

Studia Mathematica, Tome 147 (2001), p. 165-184 / Harvested from The Polish Digital Mathematics Library

Résumé

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 greedy 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.

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Publié le : 2001-01-01

Zbl 0980.41017

EUDML-ID : urn:eudml:doc:285259