Functional deconvolution in a periodic setting: uniform case. (English) Zbl 1274.62253

Pensky, Marianna; Sapatinas, Theofanis

Ann. Stat. 37, No. 1, 73-104 (2009).

Summary: 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 \(L^{2}\)-risk in the proposed functional deconvolution model when \(f(\cdot )\) 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 \(f(\cdot )\) 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.

MSC:

62G05	Nonparametric estimation
62G08	Nonparametric regression and quantile regression
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35K05	Heat equation
35L05	Wave equation

Keywords:

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

Software:

ForWaRD

References:

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