On pointwise adaptive nonparametric deconvolution

We consider estimating an unknown function f from indirect white noise obse rvations with particular emphasis on the problem of nonparametric deconvolu tion. Nonparametric estimators that can adapt to unknown smoothness of f ar e developed. The adaptive estimators are specified under two sets of assump tions on the kernel of the convolution transform. In particular, kernels ha ving Fourier transform with polynomially and exponentially decaying tails a re considered. It is shown that the proposed estimates possess, in a sense, the best possible abilities for pointwise adaptation.