Pointwise convergence of Fourier regularization for smoothing data

[摘要] The classical smoothing data problem is analyzed in a Sobolev space under the assumption of white noise. A Foufier sefies method based on regularization endowed with generalized cross validation is considered to approximate the unknown function. This approximation is globally optimal, i.e., the mean integrated squared error reaches the optimal rate in the minimax sense. In this paper the pointwise convergence property is studied. Specifically, it is proved that the smoothed solution is locally convergent but not locally optimal. Examples of functions for which the approximation is subefficient are given. It is shown that optimality and superefficiency are possible when restricting to more regular subspaces of the Sobolev space. (c) 2005 Elsevier B.V. All rights reserved.

[发布日期] 2006-11-15

[关键词] mean integrated squared error;mean squared error;smoothing data;Fourier regularization;generalized cross validation