On the Application of Linear Positive Operators for Approximation of Functions

S. B. Gashkov

Matematičeskie zametki, Tome 100 (2016) no. 5, pp. 689-700

Résumé

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 $$ t_{n,N}(f;x)=\frac{1}{N}\sum_{k=-N}^{N-1} E_n\biggl(x-\frac{\pi k}{N}\biggr)f\biggl(\frac{\pi k}{N}\biggr), $$ the Jackson-type inequalities $$ \|t_{n,N}(f;x)-f(x)\| \le (1+\pi)\omega_f\biggl(\frac1n\biggr),\qquad \|t_{n,N}(f;x)-f(x)\| \le 2\omega_f\biggl(\frac{\pi}{n+1}\biggr), $$ 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.

Détail

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