scholarly journals Approximation of classes of periodic functions of several variables with given majorant of mixed moduli of continuity

2021 ◽  
Vol 13 (3) ◽  
pp. 838-850
Author(s):  
O.V. Fedunyk-Yaremchuk ◽  
S.B. Hembars'ka
Keyword(s):  
Linear Operators ◽  
Exact Order ◽  
Moduli Of Continuity ◽  
Periodic Functions ◽  
Several Variables

In this paper, we continue the study of approximation characteristics of the classes $B^{\Omega}_{p,\theta}$ of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers. We obtain the exact-order estimates of the orthoprojective widths of the classes $B^{\Omega}_{p,\theta}$ in the space $L_{q},$ $1\leq p<q<\infty,$ and also establish the exact-order estimates of approximation for these classes of functions in the space $L_{q}$ by using linear operators satisfying certain conditions.

scholarly journals Estimates of approximative characteristics of the classes $B^{\Omega}_{p,\theta}$ of periodic functions of several variables with given majorant of mixed moduli of continuity in the space $L_{q}$

2019 ◽  
Vol 11 (2) ◽  
pp. 281-295 ◽  
Author(s):  
O.V. Fedunyk-Yaremchuk ◽  
S.B. Hembars'ka
Keyword(s):  
Linear Operators ◽  
Exact Order ◽  
Moduli Of Continuity ◽  
Periodic Functions ◽  
Several Variables

In this paper, we continue the study of approximative characteristics of the classes $B^{\Omega}_{p,\theta}$ of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers. We obtain the exact-order estimates of the orthoprojective widths of the classes $B^{\Omega}_{p,\theta}$ in the space $L_{q},$ $1\leq p<q<\infty,$ and also establish the exact-order estimates of approximation for these classes of functions in the space $L_{q}$ by using linear operators satisfying certain conditions.

Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol'skii-Besov spaces

2020 ◽  
Vol 17 (3) ◽  
pp. 372-395
Author(s):  
Anatolii Romanyuk ◽  
Viktor Romanyuk
Keyword(s):  
Besov Spaces ◽  
Best Approximation ◽  
Linear Operators ◽  
Trigonometric Polynomials ◽  
Exact Order ◽  
Periodic Functions ◽  
Sobolev Classes ◽  
The Best Approximation ◽  
Several Variables

We have obtained the exact-order estimates for some approximative characteristics of the Sobolev classes $\mathbb{W}^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}$ and Nikоl'skii--Besov classes $\mathbb{B}^{\boldsymbol{r}}_{p,\theta}\ $ of periodic functions of one and several variables in the norm of the space $B_{\infty, 1}$. Properties of the linear operators realizing the orders of the best approximation for the classes $\mathbb{B}^{\boldsymbol{r}}_{\infty, \theta}$ in this space by trigonometric polynomials generated by a set of harmonics with $``$numbers$"$ from step hyperbolic crosses are investigated.

scholarly journals Approximation characteristics of the isotropic Nikol'skii-Besov functional classes

2021 ◽  
Vol 13 (3) ◽  
pp. 851-861
Author(s):  
S.Ya. Yanchenko ◽  
O.Ya. Radchenko
Keyword(s):  
Exponential Type ◽  
Lebesgue Space ◽  
Entire Functions ◽  
Exact Order ◽  
Periodic Functions ◽  
Approximation Of Functions ◽  
Several Variables ◽  
Functions Of Exponential Type ◽  
Functional Classes ◽  
Mixed Smoothness

In the paper, we investigates the isotropic Nikol'skii-Besov classes $B^r_{p,\theta}(\mathbb{R}^d)$ of non-periodic functions of several variables, which for $d = 1$ are identical to the classes of functions with a dominant mixed smoothness $S^{r}_{p,\theta}B(\mathbb{R})$. We establish the exact-order estimates for the approximation of functions from these classes $B^r_{p,\theta}(\mathbb{R}^d)$ in the metric of the Lebesgue space $L_q(\mathbb{R}^d)$, by entire functions of exponential type with some restrictions for their spectrum in the case $1 \leqslant p \leqslant q \leqslant \infty$, $(p,q)\neq \{(1,1), (\infty, \infty)\}$, $d\geq 1$. In the case $2<p=q<\infty$, $d=1$, the established estimate is also new for the classes $S^{r}_{p,\theta}B(\mathbb{R})$.

scholarly journals Approximative characteristics of the Nikol'skii-Besov-type classes of periodic functions in the space $B_{\infty,1}$

2020 ◽  
Vol 12 (2) ◽  
pp. 376-391
Author(s):  
O.V. Fedunyk-Yaremchuk ◽  
M.V. Hembars'kyi ◽  
S.B. Hembars'ka
Keyword(s):  
Linear Operators ◽  
Partial Sums ◽  
Exact Order ◽  
Periodic Functions ◽  
Multivariate Case ◽  
The Best Approximation ◽  
Univariate Case ◽  
Type Classes ◽  
Functional Classes ◽  
Fourier Sums

We obtained the exact order estimates of the orthowidths and similar to them approximative characteristics of the Nikol'skii-Besov-type classes $B^{\Omega}_{p,\theta}$ of periodic functions of one and several variables in the space $B_{\infty,1}$. We observe, that in the multivariate case $(d\geq2)$ the orders of orthowidths of the considered functional classes are realized by their approximations by step hyperbolic Fourier sums that contain the necessary number of harmonics. In the univariate case, an optimal in the sense of order estimates for orthowidths of the corresponding functional classes there are the ordinary partial sums of their Fourier series. Besides, we note that in the univariate case the estimates of the considered approximative characteristics do not depend on the parameter $\theta$. In addition, it is established that the norms of linear operators that realize the order of the best approximation of the classes $B^{\Omega}_{p,\theta}$ in the space $B_{\infty,1}$ in the multivariate case are unbounded.

On Estimating Integral Moduli of Continuity of Functions of Several Variables in Terms of Fourier Coefficients

1999 ◽  
Vol 6 (4) ◽  
pp. 307-322
Author(s):  
L. Gogoladze
Keyword(s):  
Fourier Coefficients ◽  
Moduli Of Continuity ◽  
Several Variables

Abstract Inequalities are derived which enable one to estimate integral moduli of continuity of functions of several variables in terms of Fourier coefficients.

scholarly journals On widths of some classes of functions with values in Hilbert space

2021 ◽  
Vol 17 ◽  
pp. 105
Author(s):  
S.V. Savela
Keyword(s):  
Hilbert Space ◽  
Moduli Of Continuity ◽  
Periodic Functions

We find the exact weak widths for one class of $2\pi$-periodic functions with values in Hilbert space, which is determined by moduli of continuity.

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