Exact and asymptotic estimates for \(n\)-widths of some classes of periodic functions (Q1192588)

Let \(d_ n(A,X)\), \(d^ n(A,X)\) and \(\delta_ n(A,X)\) denote the Kolmogorov, Gel'fand and linear \(n\)-widths of \(A\) in \(X\), respectively. Let \(\tilde W^ r_ p:=\{f\mid\;f\in C^{r-1}[0,2\pi],\;f^{(i)}(0)=^{(r)}(2\pi)\) for \(i=0,1,\dots,r-1\), \(f^{(r-1)}\) absolutely continuous on \([0,2\pi]\) and \(f^{(r)}\in L^ p[0,2\pi]\}\). For the Sobolev spaces \(\tilde B^ r_ p:=\{f\mid\;f\in\tilde W^ r_ p,\;\| f^{(r)}\|_ p\leq 1\}\) the authors prove that for all \(p\in(1,\infty)\), \(d_{2n}(\tilde B^ r_ p;L^ p)=d^{2n}(\tilde B^ r_ p;L^ p)=\delta_{2n}(\tilde B^ r_ p;L^ p)=\lambda_ n\), where the numbers \(\lambda^ n\) are determined.