The Average Widths of Sobolev–Wiener Classes and Besov–Wiener Classes

Abstract

This paper concerns the problem of average σ-width of Sobolev–Wiener classes $W^{r}_{{pq}} {\left( {R^{d} } \right)},{\kern 1pt} W^{r}_{{pq}} {\left( {{\text{M}},R^{d} } \right)}$, and Besov-Wiener classes $S^{r}_{{pq\theta }} b{\left( {R^{d} } \right)},{\kern 1pt} S^{r}_{{pq\theta }} B{\left( {R^{d} } \right)},{\kern 1pt} S^{r}_{{pq\theta }} b{\left( {{\text{M}},R^{d} } \right)},{\kern 1pt} S^{r}_{{pq\theta }} B{\left( {R^{d} } \right)}$ in the metric L _q (R ^d) for 1 ≤ q ≤ p ≤ ∞. The weak asymptotic results concerning the average linear widths, the average Bernstein widths and the infinite-dimensional Gel’fand widths are obtained, respectively.