On the \(2N\)-widths of a periodic Sobolev class (Q1308785)

Let \(Q(x)\) be a polynomial with real coefficients and let \(W_ p(Q(D))\) be the Sobolev class of continuous \(2\pi\)-periodic functions \(f\) for which \(f^{d-1}\) is absolutely continuous and \(\| Q(D)f \|_ p \leq 1\), where \(d\) is the degree of \(Q\), \(D=d/dt\) and \(\| \cdot \|_ p\) is the usual \(L_ p[0,2 \pi]\)-norm. When \(Q(x)\) has only real zeros, the Kolmogorov, Gel'fand, linear, and Bernstein \(n\)-width of \(W_ p(Q(D))\) in \(L_ q [0,2\pi]\) has long been investigated by many authors. In the case when \(Q(x)\) has complex zeros, the Bernoulli function corresponding to \(Q(D)\) does not satisfy the property of cyclic variation-diminishing, and the study of these \(n\)-width becomes complicated. This question has been discussed in several recent papers. In this paper the authors obtain the exact values of \(2n\)-widths for \(p \in(1,\infty)\) and \(n>N(Q)\), where \(N(Q)\) is a constant determined simply by \(Q\). Similar results about \(n\)-widths of this paper have been proved by \textit{A. Pinkus} [Constructive Approximation 1, 15-62 (1985; Zbl 0582.41018)] for nonperiodic Sobolev classes. The proof by discretization, for \(p=q\), is complicated and does not adapt to the periodic case. The authors prove, by a variational condition, for \(p,q\) in \((0,\infty)\), the existence of the function and the eigenvalue satisfying the critical point equation.