Inverse theorems in the theory of rational approximations of functions of serveral variables
After about 2 pages of definitions and notations \(G\) open in \({\mathcal R}^ n\) with \(n\geq 2\), the Sobolev space \(W_ p^{(\ell)}\) of \(L_ p(G)\)- functions having generalized partial derivatives of order \(\ell\), isotropic modulus of smoothness, the Besov class \(B_{p,q}^{\alpha}(G)\) of functions from \(L_ p(G)\), the approximating spaces \(R_{p,q}^{\alpha}(G)\), \({\mathcal R}_{N,n}(G)\) the class of rational functions having no sigularities on \(G\) etc.) the author states and proves 4 theorems (one concerning an estimate on derivatives and three `converse' theorems about embeddings). Theorems 1-3 demand \(G\) to be a bounded, convex domain, Theorem 4 only needs \(G\) to be open and bounded. (1) \(| R|_{W_{\sigma}^{(s)}(G)}\leq C(s,p,n,G)N^ s\| R\|_{p,G}\) for \(1< p\leq\infty\), \(\sigma= (s+ 1/p)^{-1}\), \(R\in {\mathcal R}_{N,n}(G)\). (2) \(R_{\infty,1}^{\alpha}(G){\overset \subset \rightarrow} B_{1,1}^{\alpha}(G)\), \(R_{\infty,\infty}^{\alpha}(G)\) \(B_{1,\infty}^{\infty}(G)\) for \(0< \alpha< 1\). (3) If \(G\) has a locally infinitely smooth boundary, \(0< \alpha< 1\), \(1\leq q\leq\infty\), then \(R_{\infty,q}^{\alpha}(G){\overset \subset \rightarrow} B_{1,q}^{\alpha}(G)\). (4) If \(G\) is open and bounded, \(0< \alpha< 1\), \(1\leq p\leq\infty\), \(0<q\leq\infty\), then there exists for any \(\delta\), \(\delta_ 1> 0\) a closed set \(F(\delta,\delta_ 1,\alpha,p) {\overset \subset \rightarrow} G\), \(\text{mes}_ n\), \(F< C(\alpha,\delta_ 1,n)\delta\) such that \(R_{p,q}^{\alpha+ \delta_ 1}(G) {\overset \subset \rightarrow} B_{p,q}^{\alpha}(G\backslash F)\). \([\text{mes}_ n\): the \(n\)-dimensional Lebesgue measure]. The proofs of 1 and 2 are straightforward, for 3 and 4 more advanced methods are used.
scientific article; zbMATH DE number 687497