On the best polynomial approximation in Hardy space

Sharp Jackson-Stechkin-type inequalities in which the best polynomial approximation of a function in the Hardy space $H_2$ is estimated from above both in terms of the generalized modulus of continuity of the $m$-th order and in terms of the $\mathcal{K}$-functional of $r$-th derivatives are found. For some classes of functions defined with the formulated characteristics in the space $H_2$, the exact values of $n$-widths are calculated. Also in the classes $W_{2}^{(r)}(\widetilde{\omega}_{m},\Phi)$ and $W_{2}^{(r)}(\mathcal{K}_{m},\Phi)$, where $r\in\mathbb{N}$, $r\ge2$ the exact values of the best polynomial approximations of intermediate derivatives $f^{(s)}$, $1\le s\le r-1$ are obtained.