scholarly journals On approximation of continuous functions by piecewise-constant ones in integral metrics

1998 ◽  
Vol 6 ◽  
pp. 128
Author(s):  
O.V. Chernytska
Keyword(s):  
Upper Bound ◽  
Best Approximation ◽  
Continuous Functions ◽  
Moduli Of Continuity ◽  
Piecewise Constant ◽  
Integral Metrics ◽  
The Best Approximation ◽  
Piecewise Constant Functions ◽  
Approximation Of Continuous Functions ◽  
Decreasing Functions

We obtain upper bound of the best approximation of the classes $H^{\omega} [a, b]$ by piecewise-constant functions over uniform split in metrics of $L_{\varphi}[a, b]$ spaces, which are generated by continuous non-decreasing functions $\varphi$ that are equal to zero in zero. We study the classes of functions $\varphi$, for which the obtained bound is exact for all convex moduli of continuity.

scholarly journals On the nonsymmetric approximation of continuous functions in the integral metric

2020 ◽  
Vol 27 (2) ◽  
pp. 7
Author(s):  
V.F. Babenko ◽  
O.V. Polyakov
Keyword(s):  
Best Approximation ◽  
Continuous Functions ◽  
Exact Estimate ◽  
Approximation Of Continuous Functions

In the paper, an exact estimate of the best nonsymmetric approximation in the integral metric by the constants of continuous functions that belong to the classes $H^\omega[a,b]$ is proved. Taking into account Babenko's theorem on the connection of nonsymmetric approximation with the usual best approximation in the integral metric and the best one-sided approximations, from the proved result we obtain the exact estimate for the usual best approximation obtained by N.P. Korneichuk, and the exact estimate for the best one-sided approximation obtained by V.G. Doronin and A.A. Ligun.

scholarly journals Upper Estimates of the angle best approximations of generalized Liouville-Weyl derivatives

2021 ◽  
Vol 103 (3) ◽  
pp. 54-67
Author(s):  
A.E. Jetpisbayeva ◽  
◽  
A.A. Jumabayeva ◽  
Keyword(s):  
Best Approximation ◽  
Three Dimensional ◽  
Continuous Functions ◽  
Trigonometric Polynomials ◽  
Dimensional Case ◽  
Best Approximations ◽  
Approximation Of Functions ◽  
The Best Approximation

In this article we consider continuous functions f with period 2π and their approximation by trigonometric polynomials. This article is devoted to the study of estimates of the best angular approximations of generalized Liouville-Weyl derivatives by angular approximation of functions in the three-dimensional case. We consider generalized Liouville-Weyl derivatives instead of the classical mixed Weyl derivative. In choosing the issues to be considered, we followed the general approach that emerged after the work of the second author of this article. Our main goal is to prove analogs of the results of in the three-dimensional case. The concept of general monotonic sequences plays a key role in our study. Several well-known inequalities are indicated for the norms, best approximations of the r-th derivative with respect to the best approximations of the function f. The issues considered in this paper are related to the range of issues studied in the works of Bernstein. Later Stechkin and Konyushkov obtained an inequality for the best approximation f^(r). Also, in the works of Potapov, using the angle approximation, some classes of functions are considered. In subsection 1 we give the necessary notation and useful lemmas. Estimates for the norms and best approximations of the generalized Liouville-Weyl derivative in the three-dimensional case are obtained.

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