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

Abstract

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