Behaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case.
Let \(W_{0}^{1,p}(I)\) be the Sobolev space of functions with zero trace, where \(1<p<\infty\), \(I \equiv (a,b)\), \(-\infty<a<b<\infty\). Suppose that \(L^{1,p}(I)\) is the space of all functions \(u\in L^p_{loc}(I)\) with finite pseudonorm \(\| u'\| _{p,I}\) which vanishes on the subspace of all constant functions. The symbol \(L^{1,p}(I)\setminus\{1\}\) denotes a factorization of the space \(L^{1,p}(I)\) with respect to constant functions, equipped with the norm \(\| u' \| _{p,I}\). In the paper under review, it is shown that the approximation numbers \(a_{n}(E_i)\) \((i=1,2)\) of the Sobolev embeddings \[ E_1: W^{1,p}_{0}(I) \to L^p(I), \quad E_2: L^{1,p}(I)/ \{1\} \to L^{1,p}(I) / \{1\} \] have the following properties: \[ a_n(E_1)= a_n(E_2) = \frac{c_p | I| }{n} , \] where \(c_p= \frac{(p' p^{p-1})^{1/p}}{2\pi p}, \,\, \pi_p =\frac{\pi}{\sin\pi/p}. \) Besides, it is proved that \[ \inf \{ \| E_3 - p_{n+1}\| : p_{n+1}:L^{1,p}(I) \to L^p(I); \text{ rank\,} p_{n+1} < n+1\} = \frac{c_p | I| }{n} \] for the unbounded Sobolev embedding \[ E_3: L^{1,p}(I) \to L^p(I). \]
scientific article; zbMATH DE number 2057778