scholarly journals On extremal problems of approximation theory for classes of functions of two discrete variables in $l^2_{q,p}$

2021 ◽  
Vol 15 ◽  
pp. 60
Author(s):  
S.B. Vakarchuk ◽  
M.B. Vakarchuk
Keyword(s):  
Approximation Theory ◽  
Extremal Problems ◽  
Discrete Variables ◽  
Periodic Functions

For classes of $2\pi$-periodic functions of two discrete variables, defined on grid set $\sigma_{q,p}$, we found exact values of Kolmogorov and linear quasiwidths of classes $W^{r,p}(l^2_{q,p})$ in $l^2_{q,p}$ space.

Extremal problems of approximation theory in fuzzy context

1999 ◽  
Vol 105 (2) ◽  
pp. 249-257 ◽  
Author(s):  
Svetlana V. Asmuss ◽  
Alexander P. Šostak
Keyword(s):  
Approximation Theory ◽  
Extremal Problems

scholarly journals Asymptotics of approximation of functions by conjugate Poisson integrals

2020 ◽  
Vol 12 (1) ◽  
pp. 138-147
Author(s):  
I.V. Kal'chuk ◽  
Yu.I. Kharkevych ◽  
K.V. Pozharska
Keyword(s):  
Arbitrary Order ◽  
Extremal Problems ◽  
Asymptotic Distributions ◽  
Lipschitz Class ◽  
Periodic Functions ◽  
Approximation Of Functions ◽  
Poisson Integrals ◽  
Wide Range ◽  
Functional Classes ◽  
Conjugate Poisson Integrals

Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha $, i.e. the class of continuous $ 2\pi $-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0<\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha $. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.

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