PROBABILISTIC AND AVERAGE LINEAR WIDTHS OF SOBOLEV SPACE WITH GAUSSIAN MEASURE IN SPACE SQ(T) (1≤Q≤∞)

doi:10.1016/S0252-9602(15)60017-0

Acta mathematica scientia,Series B ›› 2015, Vol. 35 ›› Issue (2): 495-507.doi: 10.1016/S0252-9602(15)60017-0

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PROBABILISTIC AND AVERAGE LINEAR WIDTHS OF SOBOLEV SPACE WITH GAUSSIAN MEASURE IN SPACE S_Q(T) (1≤Q≤∞)

Yanyan XU¹, Guanggui CHEN², Ying GAN², Yan XU²

1. Lab of Security Insurance of Cyberspace of Sichuan Province, School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China;
2. School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China

Received:2014-02-19 Revised:2014-09-26 Online:2015-03-20 Published:2015-03-20
Contact: Guanggui CHEN School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China E-mail: ggchen@mail.xhu.edu.cn; 919454185@qq.com; xuyan345@163.com E-mail:ggchen@mail.xhu.edu.cn;919454185@qq.com;xuyan345@163.com
Supported by:
This work is partially supported by National Nature Science Foundation of China (61372187), Sichuan Key Technology Research and Development Program (2012GZ0019, 2013GXZ0155), and the Fund of Lab of Security Insurance of Cyberspace, Sichuan Province (szjj2014-079).

Abstract

Abstract:

Probabilistic linear (N, δ)-widths and p-average linear N-widths of Sobolev space W₂^r(T), equipped with a Gaussian probability measure μ, are studied in the metric of S_q(T) (1 ≤ q ≤ ∞), and determined the asymptotic equalities:

and

where 0< p <∞, δ∈ (0, 1/2 ], ρ >1, and S_q(T) is a subspace of S₁(T), in which the Fourier series is absolutely convergent in ?_q sense.

Key words: Probabilistic linear width, Average linear width, Gaussian measure, Sobolev space, linear operator

CLC Number:

41A46

Yanyan XU, Guanggui CHEN, Ying GAN, Yan XU. PROBABILISTIC AND AVERAGE LINEAR WIDTHS OF SOBOLEV SPACE WITH GAUSSIAN MEASURE IN SPACE S_Q(T) (1≤Q≤∞)[J].Acta mathematica scientia,Series B, 2015, 35(2): 495-507.

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PROBABILISTIC AND AVERAGE LINEAR WIDTHS OF SOBOLEV SPACE WITH GAUSSIAN MEASURE IN SPACE S_Q(T) (1≤Q≤∞)

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