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

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 $\underset{f\in W_{\beta ,1}^r}{\sup }\mathrm{\mid }f(x)-S_{n-1}(f;x)\mathrm{\mid }$ as $n\to \mathrm{\infty }$, $r\to \mathrm{\infty }$ for arbitrary relations between the parameters $r$ and $n$.