### On the number of irreducible polynomials with 0,1 coefficients

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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 $G\u2098\left(f\right):={\sum}_{k\in \Lambda}f\u0302\left(k\right){e}^{i(k,x)}$, where $\Lambda \subset {\mathbb{Z}}^{d}$ 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...

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 ${T}_{\epsilon}\left(x\right):={\sum}_{j\in {D}_{\epsilon}\left(x\right)}e{*}_{j}\left(x\right){e}_{j}$, where ${D}_{\epsilon}\left(x\right):=j:|e{*}_{j}\left(x\right)|\ge \epsilon $. We study a generalized version of ${T}_{\epsilon}$ that we call the weak thresholding approximation. We modify the ${T}_{\epsilon}\left(x\right)$ in the following way. For ε > 0, t ∈ (0,1) we set ${D}_{t,\epsilon}\left(x\right):=j:t\epsilon \le |e{*}_{j}\left(x\right)|<\epsilon $ and consider the weak...

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