Convergence of greedy approximation II. The trigonometric system

S. V. Konyagin ; V. N. Temlyakov

Studia Mathematica, Tome 157 (2003), p. 161-184 / Harvested from The Polish Digital Mathematics Library

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 $Gₘ(f) : = ∑_{k∈Λ} f̂(k)e^{i(k,x)}$, 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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Publié le : 2003-01-01

Zbl 1095.42013

EUDML-ID : urn:eudml:doc:284488