On the best polynomial approximation in the space L2 and widths of some classes of functions

Abstract

We consider the problem of the best polynomial approximation of $2\pi$-periodic functions in the space $L_2$ in the case where the error of approximation $E_{n-1}(f)$ is estimated in terms of the $k$th-order modulus of continuity $\Omega_k(f)$ in which the Steklov operator $S_h f$ is used instead of the operator of translation $T_h f (x) = f(x + h)$. For the classes of functions defined using the indicated smoothness characteristic, we determine the exact values of different $n$-widths.