Convergence of greedy approximation I. General systems
Tom 159 / 2003
Streszczenie
We consider convergence of thresholding type approximations with regard to general complete minimal systems $\{ e_n\} $ in a quasi-Banach space $X$. Thresholding approximations are defined as follows. Let $\{ e_n^*\} \subset X^*$ be the conjugate (dual) system to $\{ e_n\} $; then define for $\varepsilon >0$ and $x\in X$ the thresholding approximations as $T_\varepsilon (x) := \sum _{j\in D_\varepsilon (x)} e_j^*(x)e_j$, where $D_\varepsilon (x):= \{ j:|e_j^*(x)| \ge \varepsilon \} $. We study a generalized version of $T_\varepsilon $ that we call the weak thresholding approximation. We modify the $T_\varepsilon (x)$ in the following way. For $\varepsilon >0$, $t\in (0,1)$ we set $ D_{t,\varepsilon }(x) :=\{ j:t\varepsilon \le |e_j^*(x)|<\varepsilon \} $ and consider the weak thresholding approximations $T_{\varepsilon ,D}(x) := T_\varepsilon (x) +\sum _{j\in D} e_j^*(x)e_j$, $D\subseteq D_{t,\varepsilon }(x)$. We say that the weak thresholding approximations converge to $x$ if $T_{\varepsilon ,D(\varepsilon )}(x) \to x$ as $\varepsilon \to 0$ for any choice of $D(\varepsilon )\subseteq D_{t,\varepsilon }(x)$. We prove that the convergence set $WT\{ e_n\} $ does not depend on the parameter $t\in (0,1)$ 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.
