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A. S. Serdyuk

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

32 records · Page 2Linked to original sources
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

Lower bounds for Kolmogorov widths of classes of Poisson integrals

We expand the ranges of permissible values of $n$ ($n\in\mathbb{N}$) for which Poisson kernels $P_{q,β}(t)=\sum\limits_{k=1}^{\infty}q^k\cos\left(kt-\dfrac{βπ}{2}\right)$, ${q\in(0,1)}$, $β\in\mathbb{R}$, satisfy Kushpel's condition $C_{y,2n}$. As a consequence, we obtain exact values for Kolmogorov widths in the space $C$ ($L$) of classes $C_{β,\infty}^q$ ($C_{β,1}^q$) of Poisson integrals generated by kernels $P_{q,β}(t)$ in new situations. It is shown that obtained here results we can't obtain by using methods of finding of exact lower bounds for widths suggested by A. Pinkus.

math.CA↗
arXiv2013

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

We obtain order-exact estimates for uniform approximations by using Zygmund sums $Z^{s}_{n}$ of classes $C^ψ_{β,p}$ of $2π$-periodic continuous functions $f$ representable by convolutions of functions from unit balls of the space $L_{p}$, $1< p<\infty$, with a fixed kernels $Ψ_β\in L_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$. In addition, we find a set of allowed values of parameters (that define the class $C^ψ_{β,p}$ and the linear method $Z^{s}_{n}$) for which Zygmund sums and Fejer sums realize the order of the best uniform approximations by trigonometric polynomials of those classes.

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↗
arXiv2013

Exact values of Kolmogorov widths of classes of Poisson integrals

We prove that the Poisson kernel $P_{q,β}(t)=\sum\limits_{k=1}^{\infty}q^k\cos(kt-\dfrac{βπ}{2})$, ${q\in(0,1)}$, $β\in\mathbb{R}$, 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 $C_{β,\infty}^q$ 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 $C_{β,\infty}^q$ and show that subspaces of trigonometric polynomials of order $n-1$ are optimal for widths of dimension $2n$.

math.CA↗
arXiv2013

Order estimation of the best approximations and of the approximations by Fourier sums of classes of $(ψ,β)$--diferentiable functions

There were established the exact-order estimations of the best uniform approximations byψ the trigonometrical polynoms on the $C^ψ_{β,p}$ classes of $2π$-periodic continuous functions $f$, which are defined by the convolutions of the functions, which belong to the unit ball in $L_p$, $1\leq p <\infty$ spaces with generating fixed kernels $Ψ_β\subset|L_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$, whose Fourier coeficients decreasing to zero approximately as power functions. The exact order estimations were also established in $L_p$-metrics, $1 < p \leq\infty$ for $L^ψ_{β,1}$ classes of $2π$-periodic functions $f$, which are equivalent by means of Lebesque measure to the convolutions of $Ψ_β\subset|L_{p}$ kernels with the functions that belong to the unit ball in $L_1$ space. We showed that in investigating cases the orders of best approximations are realized by Fourier sums.

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arXiv2012

Uniform approximation of periodical functions by trigonometric sums of a special type

The approximation properties of the trigonometric sums U_{n,p}^ψof a special type are investigated on the classes C^ψ_{β, \infty} of (ψ,β)-differentiable (in the sense of Stepanets) periodical functions. The solution of Kolmogorov-Nikol'skii problem in a sufficiently general case is found as a result of consistency between the parameters of approximating sums and approximated classes. It is shown that, in some important cases the sums under consideration provide higher order of approximation in the uniform metric on the classes C^ψ_{β, \infty} than Fourier sums, Zygmund sums and de la Valle Poussin sums do. The range of parameters within the limits of it the sums U_{n,p}^ψsupply the order of the best uniform approximation on the classes C^ψ_{β, \infty} is indicated.

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↗
arXiv2012

Lebesgue-type inequalities for de la Vallee Poussin sums on the sets of analytic and entire functions

For the functions from sets $C_β^ψC$ and $C_β^ψL_s, \ 1\leq s\leq\infty$, generated by sequences $ψ(k)>0$ satisfying the condition d'Alembert $\mathop {\rm \lim}\limits_{k\rightarrow\infty}\frac{ψ(k+1)}{ψ(k)}=q, \ q\in[0,1)$, asymptotically unimprovable estimates for deviations of de la Vallée Poussin sums in the uniform metric, which are represented in terms of values of the best approximations of $(ψ,β)$-differentiable functions of this sort by trigonometric polynomials in the metrics $L_s$ are obtained. Proved that received estimates are unimprovable on some important functional subsets.

math.CA↗
arXiv2012

Approximation of continuous periodic functions by de la Vallee Poussin sums

We obtain an estimate of the deviation of de la Vallee Poussin sums V_{n,n/2}(f;x) from continuous functions f, expressed in terms of values of theirs modulus of continuity. It is established that this estimate can't be improved by using the well-known analogue of the Lebesgue inequality for de la Vallee Poussin sums

math.CA↗
arXiv2011

Approximation of classes of analytic functions by de la Vallee Poussin sums in uniform metric

In this paper asymptotic equalities are found for the least upper bounds of deviations in the uniform metric of de la Vallee Poussin sums on classes of 2π-periodic (ψ,β)-differentiable functions admitting an analytic continuation into the given strip of the complex plane. As a consequence, asymptotic equalities are obtained on classes of convolutions of periodic functions generated by the Neumann kernel and the polyharmonic Poisson kernel.

math.CA↗
arXiv2011

Uniform approximation of Poisson integrals of functions from the class H_omega by de la Vallee Poussin sums

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallée Poussin sums on the sets C^{q}_βH_ωof Poisson integrals of functions from the class H_ωgenerated by convex upwards moduli of continuity ω(t) which satisfy the condition ω(t)/t\to\infty as t\to 0. As an implication, a solution of the Kolmogorov-Nikol'skii problem for de la Vallée Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H^α, 0<α<1, is obtained

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