Asymptotic Equalities for Best Approximations for Classes of Infinitely Differentiable Functions Defined by the Modulus of Continuity

We obtain asymptotic estimates for best approximations by trigonometric polynomials in the metric of the space $C(L_p)$ for classes of periodic functions expressible as convolutions of kernels $\Psi_\beta$ with Fourier coefficients decreasing to zero faster than any power sequence, and with functions $\varphi\in C$  $(\varphi\in L_p)$ whose moduli of continuity do not exceed the given majorant of $\omega(t)$. It is proved that, in the spaces $C$ and $L_1$, for convex moduli of continuity $\omega(t)$, the obtained estimates are asymptotically sharp.