Exact estimation of the deviation of Rogosinski sums in terms of the second modulus of continuity in the space of continuous periodic functions (Q1594086)

Let \(P\) be a seminorm given on a subset \(\mathbb{M}\subset C_{2\pi}\), \(U:\mathbb{M}\rightarrow \mathbb{M}\) and \[ D(U,h)_p=\sup\left\{\frac{P(f-U(f))}{\omega_2(f,h)_p}:f\in \mathbb{M}\right\}, \] where \(w_2(f,h)_p=\sup_{|t|\leq h}P(f(\cdot +t)-2f+f(\cdot -t))\). For \[ U(f,x)=R_n(f,x)=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x+t)\left(1/2+\sum_{k=1}^n \cos\frac{k\pi}{2(n+1)}\cos kt\right)dt \] and \(D(U,h)_p\) some estimates are given. A typical result is: \(\sup\{D(R_n,h)_p:n\in\mathbb{N}\}\leq D\) and \(\lim_{n\rightarrow \infty }D(R_n,\frac{\pi}{n+1})=D\), where \(D=3/4-\frac{1}{\pi}(\text{Si}\frac{\pi}{2}- \text{Si}{\pi}+ \text{Si}\frac{3\pi}{2}+ \frac{1}{\pi^2})+\frac{1}{2\pi} \int_0^{\frac{\pi}{2}}(\frac{1}{x^2}-\frac{ctg x}{x})dx\).