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A. S. Serdyuk

Publications and source records attributed to A. S. Serdyuk.

At least 19 recordsLinked to original sources
arXiv2025

Estimates of deviations of Fourier sums on Weyl-Nagy classes $W^r_{β,1}$

We establish estimates for exact upper bounds of deviations of partial Fourier sums $S_{n-1}(f)$ on classes $W^r_{β,1}, r>2, β\in\mathbb{R},$ of $2π$-periodic functions whose $(r,β)$-derivatives in the Weyl--Nagy sense belong to the unit ball of the space $L_1$. The specified estimates allow us to write asymptotic equalities for the quantities $\sup\limits_{f\in W^r_{β,1}}|f(x)-S_{n-1}(f;x)|$ as $n\rightarrow\infty$, $r\rightarrow\infty$ for arbitrary relations between the parameters $r$ and $n$.

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arXiv2024

The best approximation by trigonometric polynomials of classes of convolutions generated by some linear combinations of periodic kernels

For arbitrary nontrivial linear combinations of a finite number of Poisson kernels, the fulfillment of the Nagy condition is established for all numbers n, starting from some number. It is also proved for any n the existence of linear combinations of m Bernoulli kernels and linear combinations of m conjugate Poisson kernels that satisfy the Nikolsky condition and at the same time do not satisfy the Nagy condition.

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arXiv2023

Estimates of approximations by interpolation trigonometric polynomials on the classes of convolutions of high smoothness

We establish interpolation analogues of Lebesgue type inequalities on the sets of $C^ψ_βL_{1}$ $2π$-periodic functions $f$, which are representable as convolutions of generating kernel $Ψ_β(t) = \sum\limits_{k=1}^{\infty}ψ(k)\cos \big(kt-\frac{βπ}{2}\big)$, $ψ(k)\geq 0$, $\sum\limits_{k=1}^{\infty}ψ(k)<\infty$, $β\in\mathbb{R}$, with functions $φ$ from $L_{1}$ . In obtained inequalities for each $x\in\mathbb{R}$ the modules of deviations $|f(x)- \tilde{S}_{n-1}(f;x)|$ of interpolation Lagrange polynomials $ \tilde{S}_{n-1}(f;\cdot)$ are estimated via best approximations $E_{n}(φ)_{L_{1}}$ of functions $φ$ by trigonometric polynomials in $L_{1}$-metrics. When the sequences $ψ(k)$ decrease to zero faster than any power function, the obtained inequalities in many important cases are asymptotically exact. In such cases we also establish the asymptotic equalities for exact upper boundaries of pointwise approximations by interpolation trigonometric polynomials on the classes of convolutions of generating kernel $Ψ_β$ with functions $φ$, which belong to the unit ball from the space $L_{1}$.

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arXiv2023

Asymptotic estimates for the widths of classes of functions of high smoothness

We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of $2π$-periodic functions $φ$, such that $\|φ\|_2\le1$, with fixed generated kernels $Ψ_{\barβ}$, which have Fourier series of the form $\sum\limits_{k=1}^\infty ψ(k)\cos(kt-β_kπ/2), $ where $ψ(k)\ge0,$ $\sumψ^2(k)<\infty, β_k\in\mathbb{R},$ in the space $C$. It is shown that for rapidly decrising sequences $ψ(k)$ (in particular, if $\lim\limits_{k\rightarrow\infty}{ψ(k+1)}/{ψ(k)}=0$) obtained estimates are asymptotic equalities. We establish that asymptotic equalities for widths of this classes are realized by trigonometric Fourier sums.

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arXiv2022

Approximation by Fourier sums in classes of Weyl--Nagy differentiable functions with high exponent of smoothness

We establish asymptotic estimates for the least upper bounds of approximations in the uniform metric by Fourier sums of order $n-1$ of classes of $2π$-periodic Weyl--Nagy differentiable functions, $W^r_{β,p}, 1\le p\le \infty, β\in\mathbb{R},$ for high exponents of smoothness $r\ (r-1\ge \sqrt{n})$. We obtain similar estimates in metrics of the spaces $L_p, 1\le p\le\infty,$ for functional classes $W^r_{β,1}$.

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arXiv2020

Asymptotic estimates for the best uniform approximations of classes of convolution of periodic functions of high smoothness

We find two-sides estimates for the best uniform approximations of classes of convolutions of $2π$-periodic functions from unit ball of the space $L_p, 1 \le p <\infty,$ with fixed kernels, modules of Fourier coefficients of which satisfy the condition $\sum\limits_{k=n+1}^\inftyψ(k)<ψ(n).$ In the case of $\sum\limits_{k=n+1}^\inftyψ(k)=o(1)ψ(n)$ the obtained estimates become the asymptotic equalities.

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arXiv2020

About Lebesgue inequalities on the classes of generalized Poisson integrals

For the functions $f$, which can be represented in the form of the convolution $f(x)=\frac{a_{0}}{2}+\frac{1}π\int\limits_{-π}^π\sum\limits_{k=1}^{\infty}e^{-αk^{r}}\cos(kt-\frac{βπ}{2})φ(x-t)dt$, $φ\perp1$, $α>0, \ r\in(0,1)$, $β\in\mathbb{R}$, we establish the Lebesgue-type inequalities of the form \begin{equation*} \|f-S_{n-1}(f)\|_{C}\leq e^{-αn^{r}}\left(\frac{4}{π^{2}}\ln \frac{n^{1-r}}{αr} + γ_{n} \right) E_{n}(φ)_{C}. \end{equation*} These inequalities take place for all numbers $n$ that are larger than some number $n_{1}=n_{1}(α,r)$, which constructively defined via parameters $α$ and $r$. We prove that there exists a function, such that the sign "$\leq$" in given estimate can be changed for "$=$".

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arXiv2020

Uniform approximations by Fourier sums on classes of convolutions of periodic functions

We establish asymptotic estimates for exact upper bounds of uniform approximations by Fourier sums on the classes of $2π$-periodic functions, which are represented by convolutions of functions $φ(φ\bot 1)$ from unit ball of the space $L_{1}$ with fixed kernels $Ψ_β$ of the form $Ψ_β(t)=\sum\limits_{k=1}^{\infty}ψ(k) \cos\left(kt-\frac{βπ}{2}\right)$, $\sum\limits_{k=1}^{\infty}kψ(k)<\infty$, $ψ(k)\geq 0$, $β\in\mathbb{R}$.

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arXiv2018

Approximation by interpolation trigonometric polynomials in metrics of the spaces $L_p$ on the classes of periodic entire functions

We obtain the asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials with the equidistant nodes $x_k^{(n-1)}=\frac{2kπ}{2n-1},\ k\in\mathbb{Z},$ in metrics of the spaces $L_p$ on classes of $2π$-periodic functions, representable as convolutions of functions $φ, \ φ\perp1,$ which belongs to the unit ball of the space $L_1$, and fixed generating kernels in the case where modules of their Fourier coefficients $ψ(k)$ satisfy the condition $\lim\limits_{k\rightarrow\infty} ψ(k+1)/ψ(k)=0.$ We obtain similar estimates on the classes of $r$-differentiable functions $W^r_1$ for the quickly increasing exponents of smoothness $r$ $(r/n\rightarrow\infty)$.

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arXiv2016

Uniform approximations by Fourier sums on classes of generalized Poisson integrals

We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of $2π$-periodic functions, representable in the form of convolutions of functions $φ$, which belong to unit balls of spaces $L_{p}$, with generalized Poisson kernels. For obtained asymptotic equalities we introduce the estimates of remainder, which are expressed in the explicit form via the parameters of the problem.

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arXiv2014

Order estimates of the best orthogonal trigonometric approximations of classes of convolutions of periodic functions of not high smoothness

We obtain order estimates for the best uniform orthogonal trigonometric approximations of $2π$-periodic functions, whose $(ψ,β)$-derivatives belong to unit balls of spaces $L_{p}, \ 1\leq p<\infty$, in case at consequences $ψ(k)$ are that product $ψ(n)n^{\frac{1}{p}}$ can tend to zero slower than any power function and $\sum\limits_{k=1}^{\infty}ψ^{p'}(k)k^{p'-2}<\infty$ when $1<p<\infty$, $\frac{1}{p}+\frac{1}{p'}=1$ and $\sum\limits_{k=1}^{\infty}ψ(k)<\infty$ when $p=1$. We also establish the analogical estimates in $L_{s}$-metric, $1< s\leq \infty$, for classes of the summable $(ψ,β)$-differentiable functions, such that $\parallel f_β^ψ\parallel_{1}\leq1$.

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arXiv2014

Estimates of best $m$-term trigonometric approximation of classes of analytic functions

In metric of spaces $L_{s}, \ 1\leq s\leq\infty$, we obtain exact in order estimates of best $m$-term trigonometric approximations of classes of convolutions of periodic functions, that belong to unit all of space $L_{p}, \ 1\leq p\leq\infty$, with generated kernel $Ψ_β(t)=\sum\limits_{k=1}^{\infty}ψ(k)\cos(kt-\frac{βπ}{2})$, $β\in \mathbb{R}$, whose coefficients $ψ(k)$ tend to zero not slower than geometric progression. Obtained estimates coincide in order with approximation by Fourier sums of the given classes of functions in $L_{s}$-metric. This fact allows to write down exact order estimates of best orthogonal trigonometric approximation and trigonometric widths of given classes.

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arXiv2014

Exact values of Kolmogorov widths of classes of analytic functions

We prove that kernels of analytic functions of kind $H_{h,β}(t)=\sum\limits_{k=1}^{\infty}\frac{1}{\cosh kh}\cos\Big(kt-\frac{βπ}{2}\Big)$, $h>0$, ${β\in\mathbb{R}}$, satisfies Kushpel's condition $C_{y,2n}$ beginning with some number $n_h$ which is explicitly expressed by parameter $h$ of smoothness of the kernel. As a consequence, for all $n\geqslant n_h$ we obtain lower bounds for Kolmogorov widths $d_{2n}$ of functional classes that are representable as convolutions of kernel $H_{h,β}$ with functions $φ\perp1$, which belong to the unit ball in the space $L_{\infty}$, in the space $C$. 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 mentioned classes of convolutions. Also for all $n\geqslant n_h$ we obtain exact values for Kolmogorov widths $d_{2n-1}$ of classes of convolutions of functions $φ\perp1$, which belong to the unit ball in the space $L_1$, with kernel $H_{h,β}$ in the space $L_1$.

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arXiv2014

Order estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in uniform metric

We obtain exact for order estimates of best uniform approximations and uniform approximations by Fourier sums of classes of convolutions the periodic functions belong to unit balls of spaces $L_{p}, \ {1\leq p<\infty}$, with generating kernel $Ψ_β$, whose absolute values of Fourier coefficients $ψ(k)$ are such that $\sum\limits_{k=1}^{\infty}ψ^{p'}(k)k^{p'-2}<\infty$, $\frac{1}{p}+\frac{1}{p'}=1$, and product $ψ(n)n^{\frac{1}{p}}$ can't tend to nought faster than power functions.

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