Order estimation of the best approximations and of the approximations by Fourier sums of classes of $(ψ,β)$--diferentiable functions

Abstract: There were established the exact-order estimations of the best uniform approximations by{\psi} the trigonometrical polynoms on the $C^{\psi}_{\beta,p}$ classes of $2\pi$-periodic continuous functions $f$, which are defined by the convolutions of the functions, which belong to the unit ball in $L_p$, $1\leq p <\infty$ spaces with generating fixed kernels $\Psi_{\beta}\subset|L_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$, whose Fourier coeficients decreasing to zero approximately as power functions. The exact order estimations were also established in $L_p$-metrics, $1 < p \leq\infty$ for $L^{\psi}_{\beta,1}$ classes of $2\pi$-periodic functions $f$, which are equivalent by means of Lebesque measure to the convolutions of $\Psi_{\beta}\subset|L_{p}$ kernels with the functions that belong to the unit ball in $L_1$ space. We showed that in investigating cases the orders of best approximations are realized by Fourier sums.