A Note on a Fixed Point Method for Deconvolution

In this paper we study a particular multidimensional deconvolution problem. The distribution of the noise is assumed to be of the form G(dx) = (1− α)δ(dx) + αg(x)dx, where δ is the Dirac mass at 0 ∈ Rd, g : Rd → [0,∞) is a density and α ∈ [0, 1 2 [. We propose a new minimax estimation procedure, which is not based on a Fourier approach, but on a fixed point method. The performances of the procedure are studied over isotropic Besov balls for Lp loss functions, 1 ≤ p <∞. A numerical study illustrates the method.