General Haar systems and greedy approximation

Anna Kamont. "General Haar systems and greedy approximation." Studia Mathematica 145.2 (2001): 165-184. <http://eudml.org/doc/285259>.

@article{AnnaKamont2001,
abstract = {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.},
author = {Anna Kamont},
journal = {Studia Mathematica},
keywords = {dyadic Haar system},
language = {eng},
number = {2},
pages = {165-184},
title = {General Haar systems and greedy approximation},
url = {http://eudml.org/doc/285259},
volume = {145},
year = {2001},
}

TY - JOUR
AU - Anna Kamont
TI - General Haar systems and greedy approximation
JO - Studia Mathematica
PY - 2001
VL - 145
IS - 2
SP - 165
EP - 184
AB - 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.
LA - eng
KW - dyadic Haar system
UR - http://eudml.org/doc/285259
ER -