Deconvolution with arbitrarily smooth kernels

We consider estimation of the density in a deconvolution model, with a smooth convoluting kernel k. For general smooth k we determine the rate and the limiting distribution for a well known Fourier kernel estimator of the density. Also, the minimax lower rate is calculated, which is attained in most cases. Remarkably, the kernel estimator, despite having the minimax optimal rate, turns out to have a degenerate limiting distribution.