Elsevier

Journal of Complexity

Volume 19, Issue 1, February 2003, Pages 73-84
Journal of Complexity

Probabilistic and average linear widths of Sobolev space with Gaussian measure

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Abstract

We determine the exact order of the p-average linear n-widths λn(a)(W2r,μ,Lq)p, 1⩽q<∞, 0<p<∞, of the Sobolev space W2r equipped with a Gaussian measure μ in the Lq-norm.

Moreover, we also calculate the probabilistic linear (n,δ)-widths and p-average linear n-widths of the finite-dimensional space Rm with the standard Gaussian measure in lqm, i.e., λn,δ(Rmm,lqm)≍m1/q−1/2m+ln(1/δ),1⩽q<2,m⩾2n,δ∈(0,1/2],λn(a)(Rmm,lqm)p≍m1/q,1⩽q<∞,0<p<∞,m⩾2n,δ∈(0,1/2].For the case of 2⩽q⩽∞, Maiorov and Wasilkowski have obtained the exact order of the probabilistic linear (n,δ)-widths λn,δ(Rmm,lqm),2⩽q⩽∞, and p-average linear n-widths λn(a)(Rmm,lqm)1,q=∞,p=1.

Keywords

Probabilistic width
Average width
Gaussian measure
Sobolev space

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Project supported by the Natural Science Foundation of China (Grant No. 10071006) and Research Fund for the Doctoral Program Higher Education.