On Jackson type inequalities for the best approximations of periodic functions (Q905375)

Given a continuous \(2\pi\)-periodic function \(f\), let \(E_n(f)\) be the best approximation of \(f\) by trigonometric polynomials of degree \(n\). Then we define \(\omega_{r,n}(f,h)=\sup_{|t|\leq h}E_n(\delta_t^r(f))\), where \(\delta_t^r(f)\) is the central difference of order \(r\) and step \(h\) for \(f\). Denote by \(\Phi_n\) the Fejer kernel. Using the integral \[ \sigma_{n,r}(f,x)=(-1)^{r+1}\frac{2\,}{C_{2r}^r}\int_{-\pi}^{\pi}\left(\sum_{k=1}^r(-1)^{k+r}C_{2r}^{k+r}f(x+kt)\right)\Phi_n(t)\,dt, \quad x\in\mathbb{R}, \] the author proves the following Jackson type inequality \[ E_n(f) \leq K(r)\,\omega_{r,n}\left(f,\frac{2\pi}{n+1}\right). \] Some inequalities related to the Jackson-Valle-Poussin kernels are also obtained.