Approximation problems on the smoothness classes

Abstract

This paper investigates the relative Kolmogorov n-widths of 2π-periodic smooth classes in \(\widetilde{L}_{q}\). We estimate the relative widths of \(\widetilde{W}^{r}H^{\omega}_{p}\) 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

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

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 ≤ ∞.