Optimal Recovery of Functions on the Sphere on a Sobolev Spaces with a Gaussian Measure in the Average Case Setting

2015 ◽  
Vol 31 (2) ◽  
pp. 154-166 ◽  
Author(s):  
Z. X. Huang and H. P. Wang
Keyword(s):  
Sobolev Spaces ◽  
Gaussian Measure ◽  
Optimal Recovery ◽  
Average Case ◽  
Average Case Setting

scholarly journals Sobolev spaces with respect to a weighted Gaussian measure in infinite dimensions

2019 ◽  
Vol 22 (04) ◽  
pp. 1950026 ◽  
Author(s):  
S. Ferrari
Keyword(s):  
Banach Space ◽  
Sobolev Spaces ◽  
Gaussian Measure ◽  
Separable Banach Space ◽  
Positive Function ◽  
Infinite Dimensions ◽  
Sobolev Function ◽  
Gradient Operator ◽  
Divergence Operator ◽  
Sobolev Functions

Let [Formula: see text] be a separable Banach space endowed with a nondegenerate centered Gaussian measure [Formula: see text] and let [Formula: see text] be a positive function on [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text] and [Formula: see text]. In this paper, we introduce and study Sobolev spaces with respect to the weighted Gaussian measure [Formula: see text]. We obtain results regarding the divergence operator (i.e. the adjoint in [Formula: see text] of the gradient operator along the Cameron–Martin space) and the trace of Sobolev functions on hypersurfaces [Formula: see text], where [Formula: see text] is a suitable version of a Sobolev function.

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