On the Application of Linear Positive Operators for Approximation of Functions

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.