Approximation of classes of periodic functions with small smoothness

The Zygmund sums of a function $f\in L_{1}$ are trigonometric polynomials of the form $$Z^{s}_{n-1}(f;t):=\frac{a_{0}}{2}+\sum_{k=1}^{n-1}\Big(1-\big(\frac{k}{n}\big)^{s}\Big) \big(a_{k}(f)\cos kt+b_{k}(f)\sin kt\big), s>0,$$ where $a_{k}(f)$ and $b_{k}(f)$ are the Fourier coefficients of $f$. We establish the exact-order estimates of uniform approximations by the Zygmund sums $Z^{s}_{n-1}$ of $2\pi$-periodic continuous functions from the classes $C^{\psi}_{\beta,p}$. These classes are defined by the convolutions of functions from the unit ball in the space $L_{p}$, $1\leq p<\infty$, with generating fixed kernels $$\Psi_{\beta}(t)\sim\sum_{k=1}^{\infty}\psi(k)\cos\left(kt+\frac{\beta\pi}{2}\right), \Psi_{\beta}\in L_{p'}, \beta\in \mathbb{R}, \frac1p+\frac{1}{p'}=1.$$ We additionally assume that the product $\psi(k)k^{s+1/p}$ is generally monotonically increasing with the rate of some power function, and, besides, for $1< p<\infty$ it holds that $\sum_{k=n}^{\infty}\psi^{p'}(k)k^{p'-2}<\infty$, and for $p=1$ the following condition $\sum_{k=n}^{\infty}\psi(k)<\infty$ is true. It is shown, that under these conditions Zygmund sums $Z^{s}_{n-1}$ and Fejér sums $\sigma_{n-1}=Z^{1}_{n-1}$ realize the order of the best uniform approximations by trigonometric polynomials of these classes, namely for $1<p<\infty$ $${E}_{n}(C^{\psi}_{\beta,p})_{C}\asymp{\cal E}\left(C^{\psi}_{\beta,p}; Z_{n-1}^{s}\right)_{C}\asymp\Big(\sum_{k=n}^{\infty}\psi^{p'}(k)k^{p'-2}\Big)^{1/p'}, \ \frac{1}{p}+\frac{1}{p'}=1,$$ and for $p=1$ $$ {E}_{n}(C^{\psi}_{\beta,1})_{C}\asymp{\cal E}\left(C^{\psi}_{\beta,1}; Z_{n-1}^{s}\right)_{C}\asymp {\left\{{\begin{array}{l l} \sum\limits_{k=n}^{\infty}\psi(k), & \cos \frac{\beta\pi}{2}\neq 0,\\ \psi(n)n, &\cos \frac{\beta\pi}{2}= 0, \end{array}} \right.} $$ where $${E}_{n}(C^{\psi}_{\beta,p})_{C}:=\sup_{f\in C^{\psi}_{\beta,p}}\inf\limits_{t_{n-1}\in\mathcal{T}_{2n-1}}\|f(\cdot)-t_{n-1}(\cdot)\|_{C}, $$ and $\mathcal{T}_{2n-1}$ is the subspace of trigonometric polynomials $t_{n-1}$ of order $n-1$ with real coefficients, $${\cal E}\left(C^{\psi}_{\beta,p}; Z_{n-1}^{s}\right)_{C}:=\mathop{\sup}\limits_{f\in C^{\psi}_{\beta,p}}\|f(\cdot)-Z^{s}_{n-1}(f;\cdot)\|_{C}.$$

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“…Serdyuk A.S. 1 , Hrabova U.Z. 2 The Zygmund sums of a function f ∈ L 1 are trigonometric polynomials of the form Z s n−1 ( f ; t) :=…”

Section: Order Estimates Of the Uniform Approximations By Zygmund Sum...mentioning

confidence: 99%

“…Approximation properties of Zygmund sums on the classes of (ψ, β)-differentiable functions were studied in the works [2,14,15], (see also [20]). Particularly in the work [2] of D.M.…”

Section: Order Estimates Of the Uniform Approximations By Zygmund Sum...mentioning

confidence: 99%

Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions

Serdyuk

Hrabova

2021

Carpathian Math. Publ.

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“…Asymptotic equalities for quantities of type (1) for various linear methods of summation of Fourier series on classes of differentiable functions of one and many variables have been obtained in [2][3][4][5][6]. We should note that in the vast majority of works, the triangular case of numerical matrices Λ has been considered.…”

Section: Introductionmentioning

confidence: 99%

Approximation of Functions of the Classes CβψHα by Linear Methods Summation of Their Fourier Series

Kharkevych,

Kal’chuk

2023

Axioms

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In this paper, we considered arbitrary linear summation methods of Fourier series specified by a set of continuous functions dependent on the real parameter and established their approximation properties. We obtained asymptotic formulas for the exact upper bounds of the deviations of operators generated by λ-methods of Fourier series summation from the functions of the classes CβψHα under certain restrictions on the functions ψ.

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“…Approximation properties of Zygmund sums on the classes of (ψ, β)-differentiable functions were studied in the works [1], [13], [14], (see., also, [19]). Particularly in the work of D.M.…”

mentioning

confidence: 99%

“…Particularly in the work of D.M. Bushev [1] the asymptotic equalities for the quantities E(C ψ β,∞ ; Z s n−1 ) C were established for some quite natural constraints on ψ and s as n → ∞. In the case, when the series ∞ k=1 ψ 2 (k) is convergent, the exact values of the quantities E C ψ β,2 ;…”

mentioning

confidence: 99%

Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions

Serdyuk¹,

Hrabova²

2020

Preprint

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Approximation of classes of periodic functions with small smoothness

Cited by 3 publications

References 2 publications

Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions

Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions

Approximation of Functions of the Classes CβψHα by Linear Methods Summation of Their Fourier Series

Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions

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