Estimates of deviations of Fourier sums on Weyl-Nagy classes $W_{\beta ,1}^r$

Abstract

We establish estimates for exact upper bounds of deviations of partial Fourier sums $S_{n-1}(f)$ on classes $W_{\beta ,1}^r,r>2,\beta \in \mathbb{R},$ of $2\pi$-periodic functions whose $(r,\beta )$-derivatives in the Weyl--Nagy sense belong to the unit ball of the space $L_1$. The specified estimates allow us to write asymptotic equalities for the quantities $\underset{f\in W_{\beta ,1}^r}{\sup }|f(x)-S_{n-1}(f;x)|$ as $n\to \infty$, $r\to \infty$ for arbitrary relations between the parameters $r$ and $n$.

10 pages