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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We consider convergence of thresholding type approximations with regard to general complete minimal systems {eₙ} in a quasi-Banach space X. Thresholding approximations are defined as follows. Let {eₙ*} ⊂ X* be the conjugate (dual) system to {eₙ}; then define for ε > 0 and x ∈ X the thresholding approximations as $T_{ε}(x) : = ∑_{j∈D_{ε}(x)} e*_{j}(x)e_{j}$, where $D_{ε}(x): = {j: |e*_{j}(x)| ≥ ε}$. We study a generalized version of $T_{ε}$ that we call the weak thresholding approximation. We modify the $T_{ε}(x)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_{t,ε}(x) : = {j: tε ≤ |e*_{j}(x)| < ε}$ and consider the weak thresholding approximations $T_{ε,D}(x) : = T_{ε}(x) + ∑_{j∈D} e*_{j}(x)e_{j}$, $D ⊆ D_{t,ε}(x)$. We say that the weak thresholding approximations converge to x if $T_{ε,D(ε)}(x) → x$ as ε → 0 for any choice of $D(ε) ⊆ D_{t,ε}(x)$. We prove that the convergence set WT{eₙ} does not depend on the parameter t ∈ (0,1) and that it is a linear set. We present some applications of general results on convergence of thresholding approximations to A-convergence of both number series and trigonometric series.
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