Pointwise convergence of Fourier regularization for smoothing data

Authors:
Daniela De Canditiis;Italia De Feis
Affiliations:
Istituto per le Applicazioni del Calcolo "Mauro Picone" (CNR), Rome, Italy;Istituto per le Applicazioni del Calcolo "Mauro Picone" (CNR), Sezione di Napoli, Naples, Italy
Venue:
Journal of Computational and Applied Mathematics
Year:
2006

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Abstract

The classical smoothing data problem is analyzed in a Sobolev space under the assumption of white noise. A Fourier series 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.