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February 2009 Functional deconvolution in a periodic setting: Uniform case
Marianna Pensky, Theofanis Sapatinas
Ann. Statist. 37(1): 73-104 (February 2009). DOI: 10.1214/07-AOS552


We extend deconvolution in a periodic setting to deal with functional data. The resulting functional deconvolution model can be viewed as a generalization of a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation. In the case when it is observed at a finite number of distinct points, the proposed functional deconvolution model can also be viewed as a multichannel deconvolution model.

We derive minimax lower bounds for the L2-risk in the proposed functional deconvolution model when f(⋅) is assumed to belong to a Besov ball and the blurring function is assumed to possess some smoothness properties, including both regular-smooth and super-smooth convolutions. Furthermore, we propose an adaptive wavelet estimator of f(⋅) that is asymptotically optimal (in the minimax sense), or near-optimal within a logarithmic factor, in a wide range of Besov balls.

In addition, we consider a discretization of the proposed functional deconvolution model and investigate when the availability of continuous data gives advantages over observations at the asymptotically large number of points. As an illustration, we discuss particular examples for both continuous and discrete settings.


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Marianna Pensky. Theofanis Sapatinas. "Functional deconvolution in a periodic setting: Uniform case." Ann. Statist. 37 (1) 73 - 104, February 2009.


Published: February 2009
First available in Project Euclid: 16 January 2009

zbMATH: 1274.62253
MathSciNet: MR2488345
Digital Object Identifier: 10.1214/07-AOS552

Primary: 62G05
Secondary: 35J05 , 35K05 , 35L05 , 62G08

Keywords: Adaptivity , Besov spaces , block thresholding , Deconvolution , Fourier analysis , functional data , Meyer wavelets , minimax estimators , multichannel deconvolution , partial differential equations , wavelet analysis

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 1 • February 2009
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