Functional Deconvolution in a Periodic Setting

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 be also viewed as a multichannel deconvolution model. We derive minimax lower bounds for the Lrisk of an estimator of the unknown response function f(·) 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. Furthermore, we propose an adaptive block thresholding 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 give advantages over observations at the asymptotically large number of points. As an illustration, we discuss particular examples for both discrete and continuous settings.