Convergence of greedy approximation II. The trigonometric system
Tom 159 / 2003
Streszczenie
We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function $f$ we take as an approximant a trigonometric polynomial of the form $G_m(f) := \sum_{k \in {\mit\Lambda}} \,\widehat{\! f}(k) e^{i(k,x)} $, where ${\mit\Lambda} \subset {\mathbb Z}^d$ is a set of cardinality $m$ containing the indices of the $m$ largest (in absolute value) Fourier coefficients $\,\widehat{\! f}(k)$ of the function $f$. Note that $G_m(f)$ gives the best $m$-term approximant in the $L_2$-norm, and therefore, for each $f\in L_2$, $\|f-G_m(f)\|_2 \to 0$ as $m\to \infty$. It is known from previous results that in the case of $p\neq 2$ the condition $f\in L_p$ does not guarantee the convergence $\|f-G_m(f)\|_p \to 0$ as $m\to \infty$. We study the following question. What conditions (in addition to $f\in L_p$) provide the convergence $\|f-G_m(f)\|_p \to 0$ as $m\to \infty$? In the case $2< p\le \infty$ we find necessary and sufficient conditions on a decreasing sequence $\{A_n\}_{n=1}^\infty$ to guarantee the $L_p$-convergence of $\{G_m(f)\}$ for all $f\in L_p$ satisfying $a_n(f)\le A_n$, where $\{a_n(f)\}$ is the decreasing rearrangement of the absolute values of the Fourier coefficients of $f$.
