On the approximation of the periodic functions by the Stechkin's polynomials (Q2880008)

The question about a lower estimate of the approximation of the periodical functions by Stechkin polynomials is investigated. It is shown that the supposition about the lower estimate mentioned above is not true. More in detail: Denote \(\| f\|=\max\limits_{x\in {\mathbb R}}| f(x)| ,\) for \(s\in {\mathbb N},\) we define \(\omega_s(f, h)=\sup\limits_{0<\delta\leq h}\| \sum\limits_{\nu=0}^s (-1)^{\nu}\left(s\atop\nu\right) f\left(\cdot+\nu\delta\right)\|,\) where \(\left(\beta\atop k\right):=\frac{\beta(\beta-1)\cdots(\beta-k+1)}{k !}.\) For \(s=2,\) the Stechkins polynomials can be written as \(\tau_{2, 8n}(f, x)=\frac{1}{2\pi\alpha_{4, n}}\int\limits_{-\pi}^{\pi} \left(2f(x+t)-f(x+2t)\right)D_n^4(t)dt,\) where \(D_n(t)=\frac{\sin\left(n+\frac{1}{2}\right)t}{\sin\frac{t}{2}}\) and \(\alpha_{4, n}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}D_n^4(t)dt.\) The main result of the paper consists in that an inequality of type \(\omega_2(f; \frac{1}{n})\leq c\| f-\tau_{2, 8n}(f)\|\) is not true with an constant \(c,\) not depening on \(f\) and \(n.\) The result mentioned above can be applied to other problems in approximation theory.