Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross
Let \(S^{r_1, \dots, r_d}_{p,p} B (\mathbb R^d)\), \(0<p<\infty\), \(r_j \in \mathbb R\) and \(S^{r_1, \dots, r_d}_p H(\mathbb R^d)\), \(1<p<\infty\), \(r_j \in \mathbb R\) be the usual Besov-Sobolev spaces with dominating mixed smoothness in \(\mathbb R^d\). It is the main aim of this paper to show that these spaces are tensor products of corresponding spaces in \(\mathbb R\), \[ S^{r_1, \dots, r_d}_{p,p}B (\mathbb R^d) = \bigotimes^d_{j=1} B^{r_j}_{p,p} (\mathbb R), \quad S^{r_1, \dots, r_d}_p H (\mathbb R^d) = \bigotimes^d_{j=1} H^{s_j}_p (\mathbb R). \] It gives the possibility to shift (orthogonal) spline wavelet bases from \(B^s_{p,p} (\mathbb R)\) and \(H^s_p (\mathbb R)\) to these spaces with dominating mixed smoothness. This includes corresponding isomorphisms to adapted sequence spaces. It is used afterwards to study best approximations related to the hyperbolic cross.
Besov-Sobolev spaces with dominating mixed smoothness
scientific article; zbMATH DE number 5655046
scientific article