Convergence of greedy approximation II. The trigonometric system

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 -