Approximations of entire functions by polynomials of the best standard approximation

Abstract

We find asymptotic equality for the top limits of approaching by the polynomials of best mean approximation on classes of $2\pi$--periodic functions $C^\psi_{\overline\beta,s}$, $1\leq s \leq\infty$, and $C^\psi_{\overline\beta}H_\omega$, that are set by multipliers $\psi(k)$ and by shifts forward an argument $\beta_k$ on condition that sequences $\psi(k)$ fall to the zero more quickly, than any geometrical progression (in this case functions from the noted classes assume regular extension upon the whole complex plane).