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 $\parallel f_{\beta}^{\psi}\parallel_{1}\leq1$.