Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure

doi:10.1016/j.jco.2004.04.001

Journal of Complexity

Volume 20, Issue 6, December 2004, Pages 858-875

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Abstract

We present sharp bounds on the Kolmogorov probabilistic (N,δ)-width and p-average N-width of multivariate Sobolev space with mixed derivative $MW_{2}^{r} (T^{d}),r=(r_{1},…,r_{d}),1/2<r_{1} =⋯=r_{ν} <r_{ν+1} ⩽⋯⩽r_{d}$ , equipped with a Gaussian measure μ in $L_{q} (T^{d})$ , that is $d_{N,δ} (MW_{2}^{r} (T^{d}),μ,L_{q} (T^{d}))≍(N^{−1} ln^{ν−1} N)^{r_{1}+(ρ−1)/2} (ln^{(ν−1)/2} N) 1+(1/N) ln (1/δ),$ $d^{(a)}_{N} (MW_{2}^{r} (T^{d}),μ,L_{q} (T^{d}))_{p} ≍(N^{−1} ln^{ν−1} N)^{r_{1}+(ρ−1)/2} (ln^{(ν−1)/2} N),$ where 1<q<∞,0<p<∞, and ρ>1 is depending only on the eigenvalues of the correlation operator of the measure μ (see (4)).

MSC

41A46

41A63

42A61

46C99

Keywords

Probabilistic width

Average width

Sobolev space with mixed derivative

Gaussian measure

Cited by (0)

^☆: Project supported by the Natural Science Foundation of China (Grant 10371009) and Research Fund for the Doctoral Program Higher Education.

Journal of Complexity

Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure☆

Abstract

MSC

Keywords