Inequalities between Best Polynomial Approximations and Some Smoothness Characteristics in the Space $L_2$ and Widths of Classes of Functions

We obtain exact constants in Jackson-type inequalities for smoothness characteristics $\Lambda_k(f)$, $k\in \mathbb{N}$, defined by averaging the $k$th-order finite differences of functions $f \in L_2$. On the basis of this, for differentiable functions in the classes $L^r_2$, $r\in \mathbb{N}$, we refine the constants in Jackson-type inequalities containing the $k$th-order modulus of continuity $\omega_k$. For classes of functions defined by their smoothness characteristics $\Lambda_k(f)$ and majorants $\Phi$ satisfying a number of conditions, we calculate the exact values of certain $n$-widths.