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

Order estimation of the best approximations and of the approximations by Fourier sums of classes of $(\psi,\beta)$--diferentiable functions