arXiv subjects

I. V. Sokolenko

Publications and source records attributed to I. V. Sokolenko.

9 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 $n$.

math.CA↗

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.

math.CA↗

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 $W^r_{β,1}$.

math.CA↗

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.

math.CA↗

arXiv2019

Approximation by Fourier sums in classes of differentiable functions with high exponents of smoothness

We find asymptotic equalities for the exact upper bounds of approximations by Fourier sums of Weyl-Nagy classes $W^r_{β,p}, 1\le p\le\infty,$ for rapidly growing exponents of smoothness $r$ $(r/n\rightarrow\infty)$ in the uniform metric. We obtain similar estimates for approximations of the classes $W^r_{β,1}$ in metrics of the spaces $L_p, 1\le p\le\infty$.

math.CA↗

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 $r$ $(r/n\rightarrow\infty)$.

math.CA↗

arXiv2017

Approximation of classes of convolutions of periodic functions by linear methods constructed on basis of Fourier-Lagrange coefficients

We calculate the least upper bounds of pointwise and uniform approximations for classes of $2π$-periodic functions expressible as convolutions of an arbitrary square summable kernel with functions, which belong to the unit ball of the space $L_2$, by linear polynomial methods constructed on the basis of their Fourier-Lagrange coefficients.

math.CA↗

arXiv2013

Approximation by linear methods of classes of $(ψ,\barβ)-$differentiable functions

We calculate the least upper bounds for approximations in the metric of the space $L_2$ by linear methods of summation of Fourier series on classes of periodic functions $L^ψ_{\barβ,1}$ defined by sequences of multipliers $ψ=ψ(k)$ and shifts of argument $\barβ=β_k$.

math.CA↗

arXiv2012

Asymptotic behavior of best approximations of classes of infinitely differentiable functions defined by moduli of continuity

We obtain asymptotic estimates for the best approximations by trigonometric polynomials in the metric space $C$ $(L_p)$ of classes of periodic functions that can be represented as a convolution of kernels $Ψ_β$, which Fourier coefficients tend to zero faster than any power sequence, with functions $φ\in C (φ\in L_p),$ which moduli of continuity do not exceed a fixed majorant $ω(t)$. It is proved that in the spaces $C$ and $L_1$ the obtained estimates are asymptotically exact for convex moduli of continuity $ω(t)$.

math.CA↗