Wyniki wyszukiwania

1

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

100%

Konyagin S.

Banach Center Publications

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2004

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tom 64

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nr 1

117-126

2

On the number of irreducible polynomials with 0,1 coefficients

100%

Konyagin S.

Acta Arithmetica

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1999

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tom 88

|

nr 4

333-350

3

On the number of polynomials of bounded measure

64%

Dubickas A., Konyagin S.

Acta Arithmetica

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1998

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tom 86

|

nr 4

325-342

4

Cyclic polygons of integer points

64%

Huxley M., Konyagin S.

Acta Arithmetica

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2009

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tom 138

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nr 2

109-136

5

Convergence of greedy approximation II. The trigonometric system

64%

Konyagin S., Temlyakov V.

Studia Mathematica

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2003

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tom 159

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nr 2

161-184

EN

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.

6

Convergence of greedy approximation I. General systems

64%

Konyagin S., Temlyakov V.

Studia Mathematica

|

2003

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tom 159

|

nr 1

143-160

EN

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.

Ograniczanie wyników

3 Acta Arithmetica

2 Studia Mathematica

1 Banach Center Publications

6 Konyagin S.

2 Temlyakov V.

1 Dubickas A.

1 Huxley M.

1 2009

1 2004

2 2003

1 1999

1 1998

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

On the number of irreducible polynomials with 0,1 coefficients

On the number of polynomials of bounded measure

Cyclic polygons of integer points

Convergence of greedy approximation II. The trigonometric system

Convergence of greedy approximation I. General systems