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

We obtain asymptotic estimates for the best approximations by trigonometric polynomials in the metric space $C$ $(L_p)$ of classes of periodic functions that can be represented as a convolution of kernels ${\mathrm{\Psi }}_{\beta }$, which Fourier coefficients tend to zero faster than any power sequence, with functions $\phi \in C(\phi \in L_p),$ which moduli of continuity do not exceed a fixed majorant $\omega (t)$. It is proved that in the spaces $C$ and $L_1$ the obtained estimates are asymptotically exact for convex moduli of continuity $\omega (t)$.