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## Studia Mathematica

2003 | 159 | 2 | 161-184
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### Convergence of greedy approximation II. The trigonometric system

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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 $Gₘ(f) : = ∑_{k∈Λ} f̂(k)e^{i(k,x)}$, 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.
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Tom
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161-184
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wydano
2003
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autor
• Department of Mathematics, Moscow State University, 119992 Moscow, Russia
autor
• Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.
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