Best m-Term Trigonometric Approximation for Periodic Functions with Low Mixed Smoothness from the Nikol’skii–Besov-Type Classes

2016 ◽  
Vol 68 (7) ◽  
pp. 1121-1145 ◽  
Author(s):  
S. A. Stasyuk
Keyword(s):  
Trigonometric Approximation ◽  
Periodic Functions ◽  
Type Classes ◽  
Mixed Smoothness

scholarly journals Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness

2015 ◽  
Vol 206 (11) ◽  
pp. 1628-1656 ◽  
Author(s):  
V N Temlyakov
Keyword(s):  
Trigonometric Approximation ◽  
Mixed Smoothness

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 $S^{r}_{p,\theta}B(\mathbb{R})$.

Approximation of functions from Nikolskii–Besov type classes of generalized mixed smoothness

2015 ◽  
Vol 41 (4) ◽  
pp. 311-334 ◽  
Author(s):  
S. A. Stasyuk ◽  
S. Ya. Yanchenko
Keyword(s):  
Approximation Of Functions ◽  
Type Classes ◽  
Generalized Mixed Smoothness ◽  
Mixed Smoothness

Approximation of periodic functions in certain sub-classes of Lp[0,2π]

2017 ◽  
Vol 10 (03) ◽  
pp. 1750046
Author(s):  
Uaday Singh ◽  
Soshal Saini
Keyword(s):  
Fourier Series ◽  
Trigonometric Approximation ◽  
Moduli Of Continuity ◽  
Periodic Functions ◽  
Matrix Means ◽  
Approximation Of Periodic Functions

In this paper, we determine the degree of trigonometric approximation of [Formula: see text]-periodic functions and their conjugates, in terms of the moduli of continuity associated with them, by matrix means of corresponding Fourier series. We also discuss some analogous results with remarks and corollaries.

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.

scholarly journals Nonlinear approximations of functions having mixed smoothness

2019 ◽  
Vol 35 (2) ◽  
pp. 119-134
Author(s):  
Cuong Manh Nguyen
Keyword(s):  
Adaptive Sampling ◽  
Finite Capacity ◽  
Entropy Numbers ◽  
Asymptotic Order ◽  
Optimal Methods ◽  
Nonlinear Approximations ◽  
Type Classes ◽  
Non Linear ◽  
Adaptive Sampling Recovery ◽  
Mixed Smoothness

For multivariate Besov-type classes $U^a_{p,\theta}$ of functions having nonuniform mixed smoothness  $a\in\rr^d_+$, we obtain the asumptotic order of entropy numbers $\epsilon_n(U^a_{p,\theta},L_q)$ and non-linear widths $\rho_n(U^a_{p,\theta},L_q)$ defined via pseudo-dimension.  We obtain also the asymptotic order of optimal methods of adaptive sampling recovery in $L_q$-norm of functions in $U^a_{p,\theta}$ by sets of a finite capacity which is measured by their cardinality or pseudo-dimension.

Entropy Numbers of the Nikol'skii—Besov-type Classes of Periodic Functions of many Variables

2019 ◽  
Vol 241 (1) ◽  
pp. 64-76
Author(s):  
Kateryna V. Pozharska
Keyword(s):  
Periodic Functions ◽  
Entropy Numbers ◽  
Type Classes
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