$\mathcal{K}$-Functionals and exact values of $n$-widths for some classes of functions in the Hardy space

In this work, we obtain exact Jackson–Stechkin type inequalities in the Hardy space $H_{q,\rho}$ ($1\le q\le\infty$, $0<\rho\le R$), in which the values of the best polynomial approximations are estimated from above in terms of the $\mathcal{K}$-functionals of the $r$th derivatives. For function classes defined by the mentioned characteristics, exact values of Bernstein and Kolmogorov $n$-widths in the space $H_{q,\rho}$ are calculated.