# Convergence of greedy approximation II. The trigonometric system

Studia Mathematica (2003)

• Volume: 159, Issue: 2, page 161-184
• ISSN: 0039-3223

top

## Abstract

top
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 $Gₘ\left(f\right):={\sum }_{k\in \Lambda }f̂\left(k\right){e}^{i\left(k,x\right)}$, where $\Lambda \subset {ℤ}^{d}$ is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in the case of p ≠ 2 the condition $f\in {L}_{p}$ does not guarantee the convergence $||f-{Gₘ\left(f\right)||}_{p}\to 0$ as m → ∞. We study the following question. What conditions (in addition to $f\in {L}_{p}$) provide the convergence $||f-{Gₘ\left(f\right)||}_{p}\to 0$ as m → ∞? In the case 2 < p ≤ ∞ we find necessary and sufficient conditions on a decreasing sequence ${Aₙ}_{n=1}^{\infty }$ to guarantee the ${L}_{p}$-convergence of Gₘ(f) for all $f\in {L}_{p}$ satisfying aₙ(f) ≤ Aₙ, where aₙ(f) is the decreasing rearrangement of the absolute values of the Fourier coefficients of f.

## How to cite

top

S. V. Konyagin, and V. N. Temlyakov. "Convergence of greedy approximation II. The trigonometric system." Studia Mathematica 159.2 (2003): 161-184. <http://eudml.org/doc/284488>.

@article{S2003,
abstract = {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 $Gₘ(f) : = ∑_\{k∈Λ\} f̂(k)e^\{i(k,x)\}$, where $Λ ⊂ ℤ^\{d\}$ is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in the case of p ≠ 2 the condition $f ∈ L_\{p\}$ does not guarantee the convergence $||f - Gₘ(f)||_\{p\} → 0$ as m → ∞. We study the following question. What conditions (in addition to $f ∈ L_\{p\}$) provide the convergence $||f - Gₘ(f)||_\{p\} → 0$ as m → ∞? In the case 2 < p ≤ ∞ we find necessary and sufficient conditions on a decreasing sequence $\{Aₙ\}_\{n=1\}^\{∞\}$ to guarantee the $L_\{p\}$-convergence of Gₘ(f) for all $f ∈ L_\{p\}$ satisfying aₙ(f) ≤ Aₙ, where aₙ(f) is the decreasing rearrangement of the absolute values of the Fourier coefficients of f.},
author = {S. V. Konyagin, V. N. Temlyakov},
journal = {Studia Mathematica},
keywords = {greedy approximation; Fourier series; Hausdorff-Young theorem; Dirichlet kernel; Rudin-Shapiro polynomial; trigonometric polynomials; decreasing rearrangement},
language = {eng},
number = {2},
pages = {161-184},
title = {Convergence of greedy approximation II. The trigonometric system},
url = {http://eudml.org/doc/284488},
volume = {159},
year = {2003},
}

TY - JOUR
AU - S. V. Konyagin
AU - V. N. Temlyakov
TI - Convergence of greedy approximation II. The trigonometric system
JO - Studia Mathematica
PY - 2003
VL - 159
IS - 2
SP - 161
EP - 184
AB - 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 $Gₘ(f) : = ∑_{k∈Λ} f̂(k)e^{i(k,x)}$, where $Λ ⊂ ℤ^{d}$ is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in the case of p ≠ 2 the condition $f ∈ L_{p}$ does not guarantee the convergence $||f - Gₘ(f)||_{p} → 0$ as m → ∞. We study the following question. What conditions (in addition to $f ∈ L_{p}$) provide the convergence $||f - Gₘ(f)||_{p} → 0$ as m → ∞? In the case 2 < p ≤ ∞ we find necessary and sufficient conditions on a decreasing sequence ${Aₙ}_{n=1}^{∞}$ to guarantee the $L_{p}$-convergence of Gₘ(f) for all $f ∈ L_{p}$ satisfying aₙ(f) ≤ Aₙ, where aₙ(f) is the decreasing rearrangement of the absolute values of the Fourier coefficients of f.
LA - eng
KW - greedy approximation; Fourier series; Hausdorff-Young theorem; Dirichlet kernel; Rudin-Shapiro polynomial; trigonometric polynomials; decreasing rearrangement
UR - http://eudml.org/doc/284488
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.