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• # Artykuł - szczegóły

## Studia Mathematica

2003 | 159 | 1 | 143-160

## Convergence of greedy approximation I. General systems

EN

### Abstrakty

EN
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 WT{eₙ} 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.

143-160

wydano
2003

### Twórcy

autor
• Department of Mechanics and Mathematics, Moscow State University, 119992 Moscow, Russia
autor
• Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.