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We find asymptotic equalities for the exact upper bounds of approximations byFourier sums of Weyl-Nagy classes $W^r_{\beta,p}, 1\le p\le\infty,$ for rapidlygrowing exponents of smoothness $r$ $(r/n\rightarrow\infty)$ in the uniformmetric. We obtain similar estimates for approximations of the classes$W^r_{\beta,1}$ in metrics of the spaces $L_p, 1\le p\le\infty$.

Approximation by Fourier sums in classes of differentiable functions with high exponents of smoothness