Convergence of greedy approximation I. General systems

S. V. Konyagin, and V. N. Temlyakov. "Convergence of greedy approximation I. General systems." Studia Mathematica 159.1 (2003): 143-160. <http://eudml.org/doc/285145>.

@article{S2003,
abstract = {We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as $T_\{ε\}(x) : = ∑_\{j∈D_\{ε\}(x)\} e*_\{j\}(x)e_\{j\}$, where $D_\{ε\}(x): = \{j: |e*_\{j\}(x)| ≥ ε\}$. We study a generalized version of $T_\{ε\}$ that we call the weak thresholding approximation. We modify the $T_\{ε\}(x)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_\{t,ε\}(x) : = \{j: tε ≤ |e*_\{j\}(x)| < ε\}$ and consider the weak thresholding approximations $T_\{ε,D\}(x) : = T_\{ε\}(x) + ∑_\{j∈D\} e*_\{j\}(x)e_\{j\}$, $D ⊆ D_\{t,ε\}(x)$. We say that the weak thresholding approximations converge to x if $T_\{ε,D(ε)\}(x) → x$ as ε → 0 for any choice of $D(ε) ⊆ D_\{t,ε\}(x)$. We prove that the convergence set WTeₙ does not depend on the parameter t ∈ (0,1) and that it is a linear set. We present some applications of general results on convergence of thresholding approximations to A-convergence of both number series and trigonometric series.},
author = {S. V. Konyagin, V. N. Temlyakov},
journal = {Studia Mathematica},
keywords = {thresholding approximations},
language = {eng},
number = {1},
pages = {143-160},
title = {Convergence of greedy approximation I. General systems},
url = {http://eudml.org/doc/285145},
volume = {159},
year = {2003},
}

TY - JOUR
AU - S. V. Konyagin
AU - V. N. Temlyakov
TI - Convergence of greedy approximation I. General systems
JO - Studia Mathematica
PY - 2003
VL - 159
IS - 1
SP - 143
EP - 160
AB - We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as $T_{ε}(x) : = ∑_{j∈D_{ε}(x)} e*_{j}(x)e_{j}$, where $D_{ε}(x): = {j: |e*_{j}(x)| ≥ ε}$. We study a generalized version of $T_{ε}$ that we call the weak thresholding approximation. We modify the $T_{ε}(x)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_{t,ε}(x) : = {j: tε ≤ |e*_{j}(x)| < ε}$ and consider the weak thresholding approximations $T_{ε,D}(x) : = T_{ε}(x) + ∑_{j∈D} e*_{j}(x)e_{j}$, $D ⊆ D_{t,ε}(x)$. We say that the weak thresholding approximations converge to x if $T_{ε,D(ε)}(x) → x$ as ε → 0 for any choice of $D(ε) ⊆ D_{t,ε}(x)$. We prove that the convergence set WTeₙ does not depend on the parameter t ∈ (0,1) and that it is a linear set. We present some applications of general results on convergence of thresholding approximations to A-convergence of both number series and trigonometric series.
LA - eng
KW - thresholding approximations
UR - http://eudml.org/doc/285145
ER -