On pointwise adaptive nonparametric deconvolution

We consider estimating an unknown function f from indirect white noise observations with particular emphasis on the problem of nonparametric deconvolution. Non-parametric estimators that can adapt to unknown smoothness of f are developed. The adaptive estimators are speciied under two sets of assumptions on the kernel of the convolution transform. In particular, kernels having the Fourier transform with polynomially and exponentially decaying tails are considered. It is shown that the proposed estimates possess, in a sense, the best possible abilities for pointwise adaptation.