The Annals of Statistics

Functional deconvolution in a periodic setting: Uniform case

Marianna Pensky and Theofanis Sapatinas

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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.

Article information

Ann. Statist., Volume 37, Number 1 (2009), 73-104.

First available in Project Euclid: 16 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G08: Nonparametric regression 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx] 35K05: Heat equation 35L05: Wave equation

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


Pensky, Marianna; Sapatinas, Theofanis. Functional deconvolution in a periodic setting: Uniform case. Ann. Statist. 37 (2009), no. 1, 73--104. doi:10.1214/07-AOS552.

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