2017
Том 69
№ 9

All Issues

Stepanyuk T. A.

Articles: 5
Article (Ukrainian)

Approximation of the classes of generalized Poisson integrals by Fourier sums in metrics of the spaces $L_s$

Serdyuk A. S., Stepanyuk T. A.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 695-704

In metrics of the spaces $L_s,\; 1 \leq s \leq \infty$, we establish asymptotic equalities for the upper bounds of approximations by Fourier sums in the classes of generalized Poisson integrals of periodic functions that belong to the unit ball of space $L_1$.

Article (Ukrainian)

Approximation of functions from the classes $W_{β}^r H^{α }$ by Weierstrass integrals

Hrabova U. Z., Kalchuk I. V., Stepanyuk T. A.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 510-519

We investigate the asymptotic behavior of the least upper bounds of the approximations of functions from the classes $W_{β}^r H^{α }$ by Weierstrass integrals in the uniform metric.

Article (Ukrainian)

Order Estimates for the Best Orthogonal Trigonometric Approximations of the Classes of Convolutions of Periodic Functions of Low Smoothness

Serdyuk A. S., Stepanyuk T. A.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 916–936

We establish order estimates for the best uniform orthogonal trigonometric approximations on the classes of $2π$-periodic functions whose $(ψ, β)$-derivatives belong to unit balls in the spaces $L_p,\; 1 ≤ p < ∞$, in the case where the sequence $ψ(k)$ is such that the product $ψ(n)n^{1/p}$ may tend to zero slower than any power function and $∑^{∞}_{k=1} ψ^{p′}(k)k^{p′−2} < ∞$ for $1 < p < ∞,\; 1\p+1\p′ = 1$, or $∑^{∞}_{k=1} ψ(k) < ∞$ for $p = 1$. Similar estimates are also established in the $L_s$-metrics, $1 < s ≤ ∞$, for the classes of summable $(ψ, β)$-differentiable functions such that $‖f_{β}^{ψ} ‖1 ≤ 1$.

Article (Ukrainian)

Order Estimates for the Best Approximations and Approximations by Fourier Sums in the Classes of Convolutions of Periodic Functions of Low Smoothness in the Uniform Metric

Serdyuk A. S., Stepanyuk T. A.

↓ Abstract

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1658–1675

We obtain the exact-order estimates for the best uniform approximations and uniform approximations by Fourier sums in the classes of convolutions of periodic functions from the unit balls of the spaces $L_p, 1 ≤ p < ∞$, with generating kernel $Ψ_{β}$ for which the absolute values of its Fourier coefficients $ψ(k)$ are such that $∑_{k = 1}^{∞} ψ_p ′(k)k^{p ′ − 2} < ∞,\; \frac 1p + \frac 1{p′} = 1$, and the product $ψ(n)n^{1/p}$ cannot tend to zero faster than power functions.

Article (Ukrainian)

Estimations of the Best Approximations for the Classes of Infinitely Differentiable Functions in Uniform and Integral Metrics

Serdyuk A. S., Stepanyuk T. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1244–1256

We establish uniform (with respect to the parameter p, 1 ≤ p ≤ ∞) upper estimations of the best approximations by trigonometric polynomials for the classes C β,p ψ of periodic functions generated by sequences ψ(k) vanishing faster than any power function. The obtained estimations are exact in order and contain constants expressed in the explicit form and depending solely on the function ψ. Similar estimations are obtained for the best approximations of the classes L β,1 ψ in metrics of the spaces L s , 1 ≤ s ≤ ∞.