# Convergence of greedy approximation II. The trigonometric system

Studia Mathematica (2003)

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

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## Abstract

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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.

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