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T. A. Stepaniuk

Publications and source records attributed to T. A. Stepaniuk.

10 recordsLinked to original sources
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 $L_{1}$.

math.CA↗
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 "$=$".

math.CA↗
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.

math.CA↗
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 $\parallel f_β^ψ\parallel_{1}\leq1$.

math.CA↗
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.

math.CA↗
arXiv2014

Estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in integral metrics

In metric of spaces $L_{s}, \ 1< s\leq\infty$, we obtain exact order estimates of best approximations and approximations by Fourier sums of classes of convolutions the periodic functions that belong to unit ball of space $L_{1}$, with generating kernel $Ψ_β(t)=\sum\limits_{k=1}^{\infty}ψ(k)\cos(kt-\frac{βπ}{2})$, $β\in\mathbb{R}$, whose coefficients $ψ(k)$ are such that product $ψ(n)n^{1-\frac{1}{s}}$, $1<s\leq\infty$, can't tend to nought faster than every power function and besides, if $1<s<\infty$, then $\sum\limits_{k=1}^{\infty}ψ^{s}(k)k^{s-2}<\infty$ and if $s=\infty$, then $\sum\limits_{k=1}^{\infty}ψ(k)<\infty$.

math.CA↗
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.

math.CA↗
arXiv2013

Estimations of the best approximations of classes of infinitely differentiable functions in uniform and integral metrics

We find uniform with respect to parameter $p \ (1\leq p\leq\infty)$ upper estimations of best approximations by trigonometric polynomials of classes $C^ψ_{β,p}$ of periodic functions generated by sequences $ψ(k)$, that decrease to nought faster than any power function. Obtained estimations are exact for order and have constants that are written in explicit form and depend on function $ψ$ only. We obtain analogical estimations for best approximations of classes $L^ψ_{β,1}$ in metrics of spaces $L_{s}$, $1\leq s\leq \infty$.

math.CA↗
arXiv2013

Order estimations of the best approximations and approximations of the Fourier sums on the classes of infinitely differentiable functions

We obtained order estimations for the best uniform approximations by trigonometric polynomials and approximations by Fourier sums of classes of $2π$-periodic continuous functions, which $(ψ,β)$-derivatives $f_β^ψ$ belong to unit balls of spaces $L_{p}, 1\leq p<\infty$ in case at consequences $ψ(k)$ decrease to nought faster than any power function. We also established the analogical estimations in $L_{s}$-metric, $1<s\leq\infty$, for classes of the summable $(ψ,β)$-differentiable functions, such that $\parallel f_β^ψ\parallel_{1}\leq1$.

math.CA↗