  We establish asymptotic estimates for the least upper bounds of
approximations in the uniform metric by Fourier sums of order $n-1$ of classes
of $2\pi$-periodic Weyl--Nagy differentiable functions, $W^r_{\beta,p}, 1\le
p\le \infty, \beta\in\mathbb{R},$ for high exponents of smoothness $r\ (r-1\ge
\sqrt{n})$. We obtain similar estimates in metrics of the spaces $L_p, 1\le
p\le\infty,$ for functional classes $W^r_{\beta,1}$.
