On the application of linear positive operators for approximation of functions

Abstract

For the linear positive Korovkin operator $f\left( x \right) \to {t_n}\left( {f;x} \right) = \frac{1}{\pi }\int_{ - \pi }^\pi {f\left( {x + t} \right)E\left( t \right)dt}$, where E(x) is the Egervary–Szász polynomial and the corresponding interpolation mean ${t_{n,N}}\left( {f;x} \right) = \frac{1}{N}\sum\limits_{k = - N}^{N - 1} {{E_n}\left( {x - \frac{{\pi k}}{N}} \right)f\left( {\frac{{\pi k}}{N}} \right)}$ , the Jackson-type inequalities $\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant \left( {1 + \pi } \right){\omega _f}\left( {\frac{1}{n}} \right),\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant 2{\omega _f}\left( {\frac{\pi }{{n + 1}}} \right)$, where ωf (x) denotes the modulus of continuity, are proved for N > n/2. For ωf (x) ≤ Mx, the inequality $\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant \frac{{\pi M}}{{n + 1}}$. 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.