Approximation problems on the smoothness classes

This paper investigates the relative Kolmogorov n-widths of 2π-periodic smooth classes in \(\widetilde{L}_{q}\) $${\tilde{L}}_q$$ . We estimate the relative widths of \(\widetilde{W}^{r}H^{\omega}_{p}\) $${\tilde{W}}^r{H}_p^{\omega }$$ and its generalized class K_pH^ω (P_r), where K_pH^ω (P_r) is defined by a self-conjugate differential operator P_r (D) induced by

\(P_{r}(t):= t^{\sigma} \Pi_{j=1}^{l}(t^{2}- t_{j}^{2}),~t_{j} > 0,~j=1, 2,\cdots, l,~l \geq 1,~\sigma \geq 1,~r=2l+\sigma.\) $$P_r(t):=t^{\sigma }{\mathrm{\Pi }}_{j=1}^l(t^2-t_j^2),t_j>0,j=1,2,\cdots ,l,l\ge 1,\sigma \ge 1,r=2l+\sigma .$$

Also, the modulus of continuity of the r-th derivative, or r-th self-conjugate differential, does not exceed a given modulus of continuity ω. Then we obtain the asymptotic results, especially for the case p = ∞, 1 ≤ q ≤ ∞.