Greedy approximation and the multivariate Haar system

A. Kamont, and V. N. Temlyakov. "Greedy approximation and the multivariate Haar system." Studia Mathematica 161.3 (2004): 199-223. <http://eudml.org/doc/285349>.

@article{A2004,
abstract = {We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis in $L_\{p\}([0,1])$, 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis $^\{d\} = × ... × $ in $L_\{p\}([0,1]^\{d\})$, 1 < p < ∞, p ≠ 2). We prove some convergence results and also some results on convergence rate of weak type greedy algorithms. Our results are expressed in terms of properties of the basis with respect to a given weakness sequence.},
author = {A. Kamont, V. N. Temlyakov},
journal = {Studia Mathematica},
keywords = {greedy basis; greedy algorithm; Haar basis; multivariate Haar basis; weak thresholding; Besove class},
language = {eng},
number = {3},
pages = {199-223},
title = {Greedy approximation and the multivariate Haar system},
url = {http://eudml.org/doc/285349},
volume = {161},
year = {2004},
}

TY - JOUR
AU - A. Kamont
AU - V. N. Temlyakov
TI - Greedy approximation and the multivariate Haar system
JO - Studia Mathematica
PY - 2004
VL - 161
IS - 3
SP - 199
EP - 223
AB - We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis in $L_{p}([0,1])$, 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis $^{d} = × ... × $ in $L_{p}([0,1]^{d})$, 1 < p < ∞, p ≠ 2). We prove some convergence results and also some results on convergence rate of weak type greedy algorithms. Our results are expressed in terms of properties of the basis with respect to a given weakness sequence.
LA - eng
KW - greedy basis; greedy algorithm; Haar basis; multivariate Haar basis; weak thresholding; Besove class
UR - http://eudml.org/doc/285349
ER -