Exact and asymptotic estimates forn-widths of some classes of periodic functions

Abstract

Let$\tilde W_p^r : = \left\{ {f\left| {f \in C^{r - 1} } \right.} \right.\left[ {0,2\pi } \right],f^{(i)} (0) = f^{(i)} (2\pi ),i = 0, \ldots ,r - 1,f^{(r - 1)}$, abs. cont. on [0, 2π] andf ^(r)∈L ^p[0, 2π]}, and set$\tilde B_p^r : = \left\{ {f\left| {f \in \tilde W_p^r ,} \right.\left\| {f^{(r)} } \right\|_p \leqslant 1} \right\}$. We find the exact Kolmogrov, Gel'fand, and linearn-widths of$\tilde B_p^r$ inL ^p forn even and allp∈(1, ∞). The strong asymptotic estimates forn-widths of$\tilde B_p^r$ inL ^p are also obtained.