Optimal recovery of functions on the sphere on a Sobolev spaces with a Gaussian measure in the average case setting

HUANG Zexia; WANG He-Ping

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Optimal recovery of functions on the sphere on a Sobolev spaces with a Gaussian measure in the average case setting

HUANG Zexia,WANG He-Ping *

School of Mathematical Sciences, Capital Normal University, Beijing 100048

*Correspondence author

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Funding: the Beijing Natural Science Foundation （No.1102011）, Specialized Research Fund for the Doctoral Program of Higher Education（No.20091108110004）, Supported by the National Natural Science Foundation of China（No.Projectno.10871132,11271263）

Opened online:17 January 2013

Accepted by: none

Citation: HUANG Zexia,WANG He-Ping.Optimal recovery of functions on the sphere on a Sobolev spaces with a Gaussian measure in the average case setting[OL]. [17 January 2013] http://en.paper.edu.cn/en_releasepaper/content/4514337

Optimal recovery means that using finitely manyarbitrary function values f(x) for some x∈D to reconstruct(recovery) functions f from a given classes with the least possible errors. Optimal recovery constitutes a important ingredient innumerical analysis and has many important practical applications.There are two most important case setting: worst case setting andaverage case setting as far as error measure is concerned. In thispaper, optimal recovery (reconstruction) of functions on the spherein the average case setting is studied. The asymptotic orders ofaverage sampling numbers of a Sobolev space on the sphere with aGaussian measure in the Lq(sd-1) metric for 1≤q≤∞ are obtained, and it is shown that some worst-caseasymptotically optimal algorithms are also asymptotically optimal in the average case setting inthe Lq(sd-1) metric for 1≤q≤∞.

Keywords:Approximation of functions; Optimal recovery; averagesampling numbers; optimal algorithm; Gaussian measure.

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