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Acta Arithmetica

### Convergent subsequences of partial sums of Fourier series of ϕ(L)

Banach Center Publications

Acta Arithmetica

### Convergence of greedy approximation II. The trigonometric system

Studia Mathematica

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...

### Convergence of greedy approximation I. General systems

Studia Mathematica

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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