scholarly journals Approximation of the Nikol'skii-Besov functional classes by entire functions of a special form

2020 ◽  
Vol 12 (1) ◽  
pp. 148-156
Author(s):  
S.Ya. Yanchenko
Keyword(s):  
Exponential Type ◽  
Lebesgue Space ◽  
Special Form ◽  
Entire Functions ◽  
Exact Order ◽  
Approximation Of Functions ◽  
Functions Of Exponential Type ◽  
Functional Classes

We establish the exact-order estimates for the approximation of functions from the Nikol'skii-Besov classes $S^{\boldsymbol{r}}_{1,\theta}B(\mathbb{R}^d)$, $d\geq 1$, by entire functions of exponential type with some restrictions for their spectrum. The error of the approximation is estimated in the metric of the Lebesgue space $L_{\infty}(\mathbb{R}^d)$.

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 The use of the isometry of function spaces with different numbers of variables in the theory of approximation of functions

2021 ◽  
Vol 13 (3) ◽  
pp. 805-817
Author(s):  
D.M. Bushev ◽  
F.G. Abdullayev ◽  
I.V. Kal'chuk ◽  
M. Imashkyzy
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Integral Representations ◽  
Periodic Functions ◽  
Approximation Of Functions ◽  
Functions Of Exponential Type

In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of the delta-like kernels used to construct isometric function spaces with various numbers of variables.

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