Probabilistic and average linear widths of Sobolev space with Gaussian measure

✍ Scribed by Fang Gensun; Ye Peixin

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 155 KB
• Volume: 19
• Category: Article
• ISSN: 0885-064X
• DOI: 10.1016/s0885-064x(02)00005-5

✦ Synopsis

We determine the exact order of the p-average linear n-widths l ðaÞ n ðW r 2 ; m; L q Þ p ; 1pqoN; 0opoN; of the Sobolev space W r 2 equipped with a Gaussian measure m in the L q -norm. Moreover, we also calculate the probabilistic linear ðn; dÞ-widths and p-average linear nwidths of the finite-dimensional space R m with the standard Gaussian measure in l m q ; i.e., l n;d ðR m ; n m ; l m q Þ^m 1=qÀ1=2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi m þ lnð1=dÞ p ; 1pqo2; mX2n; dAð0; 1=2; l ðaÞ n ðR m ; n m ; l m q Þ p ^m1=q ; 1pqoN; 0opoN; mX2n; dAð0; 1=2:

For the case of 2pqpN; Maiorov and Wasilkowski have obtained the exact order of the probabilistic linear ðn; dÞ-widths l n;d ðR m ; n m ; l m q Þ; 2pqpN; and p-average linear n-widths l ðaÞ n ðR m ; n m ; l m q Þ 1 ; q ¼ N; p ¼