Convergence of greedy approximation II. The trigonometric system

S. V. Konyagin¹ ; V. N. Temlyakov²

¹ Department of Mathematics Moscow State University 119992 Moscow, Russia
² Department of Mathematics University of South Carolina Columbia, SC 29208, U.S.A.

Studia Mathematica, Tome 159 (2003) no. 2, pp. 161-184

Résumé

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 $f$.

Détail

DOI : 10.4064/sm159-2-1

Keywords: study following nonlinear method approximation trigonometric polynomials periodic function approximant trigonometric polynomial form sum mit lambda widehat where mit lambda subset mathbb set cardinality containing indices largest absolute value fourier coefficients widehat function note gives best m term approximant norm therefore each f g infty known previous results neq condition does guarantee convergence f g infty study following question what conditions addition provide convergence f g infty infty necessary sufficient conditions decreasing sequence infty guarantee p convergence satisfying where decreasing rearrangement absolute values fourier coefficients