Widths of classes of convolutions with Poisson kernel

Shevaldin, V. T.

doi:10.1007/bf01263308

Cited by 8 publications

(7 citation statements)

References 4 publications

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“…• For any n ∈ N and β ∈ R with 0 < q ≤ q(β) where q(β) = 0.2 for β ∈ Z or q(β) = 0.196881 for β ∈ R \ Z (see [17]);…”

Section: Introductionmentioning

confidence: 99%

“…Note that the problem of finding exact values of widths of classes of Poisson integrals C q β,p , p = 1, ∞, is considered in [5][6][7]17]. As for the widths of sets of Poisson integrals for functions from classes H ω that are generated by the module of continuity, sharp or asymptotically sharp estimates are obtained only in certain cases (see e.g.…”

mentioning

confidence: 99%

“…. Sufficient conditions of the inclusion Ψ β ∈ C y,2n for kernels of the form (1) were suggested in [6,7,10,12,14,17,19]. By using these conditions, these authors succeeded to apply Theorem 1 and obtain sharp estimates for widths d m (C ψ β,∞ , C) and d m (C ψ β,1 , L) in several new cases.…”

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confidence: 99%

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Exact values of Kolmogorov widths of classes of Poisson integrals

Serdyuk

Bodenchuk

2013

Journal of Approximation Theory

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We prove that the Poisson kernel $P_{q,\beta}(t)=\sum\limits_{k=1}^{\infty}q^k\cos(kt-\dfrac{\beta\pi}{2})$, ${q\in(0,1)}$, $\beta\in\mathbb{R}$, satisfies Kushpel's condition $C_{y,2n}$ beginning with a number $n_q$ where $n_q$ is the smallest number $n\geq9$, for which the following inequality is satisfied: $$ \dfrac{43}{10(1-q)}q^{\sqrt{n}}+\dfrac{160}{57(n-\sqrt{n})}\; \dfrac{q}{(1-q)^2}\leq (\dfrac{1}{2}+\dfrac{2q}{(1+q^2)(1-q)})(\dfrac{1-q}{1+q})^{\frac {4}{1-q^2}}. $$ As a consequence, for all $n\geq n_q$ we obtain lower bounds for Kolmogorov widths in the space $C$ of classes $C_{\beta,\infty}^q$ of Poisson integrals of functions that belong to the unit ball in the space $L_\infty$. The obtained estimates coincide with the best uniform approximations by trigonometric polynomials for these classes. As a result, we obtain exact values for widths of classes $C_{\beta,\infty}^q$ and show that subspaces of trigonometric polynomials of order $n-1$ are optimal for widths of dimension $2n$

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“…• For any n ∈ N and β ∈ R with 0 < q ≤ q(β) where q(β) = 0.2 for β ∈ Z or q(β) = 0.196881 for β ∈ R \ Z (see [17]);…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Exact values of Kolmogorov widths of classes of Poisson integrals

Serdyuk

Bodenchuk

2013

Journal of Approximation Theory

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“…In particular, it was proved in [35,36] that, for β ∈ Z, the condition C y n ,2 , n ∈ N , is satisfied by functions Ψ β ( ) t of the form (55) with coefficients ψ ( k ) = ϕ ρ ( ) k k , 1 < ρ ≤ 1 / 7 , where ϕ ( k ) are arbitrary positive nonincreasing functions of natural argument. Later, Shevaldin proved in [37,38] that the condition C y n It was proved in [20] that the conditions C y n ,2 , n ∈ N , are also satisfied by kernels Ψ β ( ) t of the form (5) whose coefficients ψ ( k ) satisfy the inequalities…”

Section: Estimates For Kolmogorov Widths Of Classesmentioning

confidence: 97%

“…The final results concerning the solution of the problem of finding the exact values of quantities (6) and (7) in the case ψ ( ) k k r = − for arbitrary r > 0 and β β k ≡ , β ∈ R , were obtained by Dzyadyk in [9]. For ψ ( ) k q k = , 0 < q < 1, i.e., in the case where the functional classes C β ψ ,∞ and L β ψ ,1 are generated by Poisson kernels P t q, ( ) , 0 < q < 1, β ∈ R, the exact values of quantities (6) and (7) were determined by Krein [15] and Nikol'skii [6] (in the metrics of C and L, respectively, β ∈ Z ), Bushanskii [16], and Shevaldin [17] ( ) β ∈R . In the case where the classes C β ψ ,∞ and L β ψ ,1 are generated by even kernels Ψ 0 ( ) t of the form…”

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confidence: 99%

Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

Serdyuk

2005

Ukr Math J

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We consider classes of 2π-periodic functions that are represented in terms of convolutions with fixed kernels Ψ β whose Fourier coefficients tend to zero at exponential rate. We determine exact values of the best approximations of these classes in the uniform and integral metrics. In several cases, we determine the exact values of the Kolmogorov, Bernstein, and linear widths for these classes in the metrics of the spaces C and L.Let L = L 1 be the space of 2π-periodic functions ϕ summable on [ 0, 2 π ] with the normthe space of measurable, essentially bounded, 2π-periodic functions ϕ with the norm ϕ ∞ = ess sup t t ϕ ( ) , and let C be the space of continuous 2π-periodic functions ϕ with the norm ϕ C = max t t ϕ ( ) .In the present paper, we use classes of functions introduced by Stepanets [1], namely, the classes L β ψ ᑨ,

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Asymptotic Estimates for the Best Uniform Approximations of Classes of Convolution of Periodic Functions of High Smoothness

Serdyuk

Sokolenko

2021

J Math Sci

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Widths of classes of convolutions with Poisson kernel

Cited by 8 publications

References 4 publications

Exact values of Kolmogorov widths of classes of Poisson integrals

Exact values of Kolmogorov widths of classes of Poisson integrals

Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

Asymptotic Estimates for the Best Uniform Approximations of Classes of Convolution of Periodic Functions of High Smoothness

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