Estimates of deviations of Fourier sums on Weyl-Nagy classes $W^r_{β,1}$

We establish estimates for exact upper bounds of deviations of partial Fourier sums $S_{n-1}(f)$ on classes $W^r_{β,1}, r>2, β\in\mathbb{R},$ of $2π$-periodic functions whose $(r,β)$-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 $\sup\limits_{f\in W^r_{β,1}}|f(x)-S_{n-1}(f;x)|$ as $n\rightarrow\infty$, $r\rightarrow\infty$ for arbitrary relations between the parameters $r$ and $n$.