On the \(2n\)-widths of a periodic Sobolev class
Let \(Q(x)\) be a polynomial with real coefficients and the Sobolev class \(W_ p(Q(D))\) be the set of continuous \(2\pi\)-periodic functions \(f(x)\) for which \(f^{(\deg Q-1)}\) is absolutely continuous and \(\| Q(D) f\|_ p\leq 1\), where \(\deg Q\) is the degree of \(Q\), \(D= {d\over {dt}}\), and \(\|\cdot\|_ p\) is the usual \(L_ p[0,2\pi]\) norm. We denote by \(d_ n(p,q)\), \(d^ n(p,q)\), \(\delta_ n(p,q)\) and \(b_ n(p,q)\) the Kolmogorov, Gelfand, linear and Bernstein \(n\)-widths of \(W_ p(Q(D))\) in \(L_ q[0,2\pi]\) respectively. The authors obtain the exact values of \(d_{2n}(p,p)\), \(d^{2n}(p,p)\) and \(\delta_{2n}(p,p)\) for \(p\in (1,\infty)\) and \(n>N(Q)\), where \(N(Q)\) is a constant determined simply by \(Q\).
scientific article; zbMATH DE number 177996