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 Mat. Zametki, 2016, Volume 100, Issue 5, Pages 689–700 (Mi mz11064)

On the Application of Linear Positive Operators for Approximation of Functions

S. B. Gashkov

Lomonosov Moscow State University

Abstract: For the linear positive Korovkin operator
$$f(x)\to t_n(f;x)=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x+t)E(t) dt,$$
where $E(x)$ is the Egervary–Szász polynomial and the corresponding interpolation mean
$$t_{n,N}(f;x)=\frac{1}{N}\sum_{k=-N}^{N-1} E_n(x-\frac{\pi k}{N})f(\frac{\pi k}{N}),$$
the Jackson-type inequalities
$$\|t_{n,N}(f;x)-f(x)\| \le (1+\pi)\omega_f(\frac1n),\qquad \|t_{n,N}(f;x)-f(x)\| \le 2\omega_f(\frac{\pi}{n+1}),$$
where $\omega_f(x)$ denotes the modulus of continuity, are proved for $N > n/2$. For $\omega_f(x) \le Mx$, the inequality
$$\|t_{n,N}(f;x)-f(x)\| \le \frac{\pi M}{n+1} \mspace{2mu}.$$
is established. As a consequence, an elementary derivation of an asymptotically sharp estimate of the Kolmogorov width of a compact set of functions satisfying the Lipschitz condition is obtained.

Keywords: positive linear operators, Korovkin operator, interpolation mean, trigonometric polynomial, Egervary–Szász polynomial, Jackson-type inequality, functions satisfying the Lipschitz condition, Kolmogorov width.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-00598 This work was supported by the Russian Foundation for Basic Research under grant 14-01-00598.

DOI: https://doi.org/10.4213/mzm11064

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English version:
Mathematical Notes, 2016, 100:5, 666–676

Bibliographic databases:

UDC: 517.518.8
Revised: 08.05.2016

Citation: S. B. Gashkov, “On the Application of Linear Positive Operators for Approximation of Functions”, Mat. Zametki, 100:5 (2016), 689–700; Math. Notes, 100:5 (2016), 666–676

Citation in format AMSBIB
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