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We obtain asymptotic estimates for the best approximations by trigonometricpolynomials in the metric space $C$ $(L_p)$ of classes of periodic functionsthat can be represented as a convolution of kernels $\Psi_\beta$, which Fouriercoefficients tend to zero faster than any power sequence, with functions$\varphi\in C (\varphi\in L_p),$ which moduli of continuity do not exceed afixed majorant $\omega(t)$. It is proved that in the spaces $C$ and $L_1$ theobtained estimates are asymptotically exact for convex moduli of continuity$\omega(t)$.

Asymptotic behavior of best approximations of classes of infinitely differentiable functions defined by moduli of continuity