Order estimations of the best approximations and approximations of the Fourier sums on the classes of infinitely differentiable functions

Abstract: We obtained order estimations for the best uniform approximations by trigonometric polynomials and approximations by Fourier sums of classes of $2\pi$-periodic continuous functions, which $(\psi,\beta)$-derivatives $f_{\beta}^{\psi}$ belong to unit balls of spaces $L_{p}, 1\leq p<\infty$ in case at consequences $\psi(k)$ decrease to nought faster than any power function. We also established the analogical estimations in $L_{s}$-metric, $1<s\leq\infty$, for classes of the summable $(\psi,\beta)$-differentiable functions, such that $\parallel f_{\beta}^{\psi}\parallel_{1}\leq1$.