Asymptotic estimates of approximation of continuous periodic functions by Fourier sums (Q1813756)

The author proves the asymptotic estimates expressed by the modulus of continuity of \(r\)-th order \((r\geq 2)\) of \(f\in C_{2\pi}\) or \((\psi,\beta)\)-derivative of \(f\in C^ \psi_ \beta C\). One of them is given in the following Theorem 1. For any \(f\in C_{2\pi}\) such that \(\omega_ r(f,\pi/n)>0\) \((r\geq 2)\) the following estimates \[ \| f- S_{n-1}(f)\|_ C < A_ r(\pi/n) \omega_ r(f,\pi/n), \] \[ A_ r(\pi/n):=\sup_{{f\in C_{2\pi}\atop \omega_ r(f,\pi/n)>0}}\{\| f-S_{n-1}(f)\|_ C/\omega_ r(f,\pi/n)\}={2^{2-r}\over \pi^ 2}\ln n+O(1) \] hold.