Widths between the anisotropic spaces and the spaces of functions with mixed smoothness
The main focus of the paper is on the derivation of sharp asymptotic orders of the Kolmogorov, Gelfand, linear, and orthoprojection widths of the classes \(BF^r_{\text{mix}, p}\) in the anisotropic space \(F^{\mathbf{R}}_q\) and those of the classes \(BF^{\mathbf {R}}_q\) in the space \(F^r_{\text{mix},q}\), for all \(p,q\in [1, \infty]\). Here, \(F^r_{\text{mix},q}\) denotes either the Sobolev space \(W^r_{\text{mix},q}\) of mixed smoothness or the Hölder-Nikolskii space \(H^r_{\text{mix},q}\) of mixed smoothness, and \(F^{\mathbf {R}}_q\) either the anisotropic Sobolev space \(W^{\mathbf{R}}_q\) or the anisotropic Hölder-Nikolskij space \(H^{\mathbf{R}}_q\). The quantities \(BF^{\mathbf {R}}_q\) and \(BF^{\mathbf {R}}_q\) represent the unit ball of the spaces \(F^r_{\text{mix},q}\) and \(F^{\mathbf {R}}_q\), respectively.
spaces of functions with mixed smoothness
scientific article; zbMATH DE number 6009280