  In this paper we study the problem of density deconvolution under general
assumptions on the measurement error distribution. Typically deconvolution
estimators are constructed using Fourier transform techniques, and it is
assumed that the characteristic function of the measurement errors does not
have zeros on the real line. This assumption is rather strong and is not
fulfilled in many cases of interest. In this paper we develop a methodology for
constructing optimal density deconvolution estimators in the general setting
that covers vanishing and non--vanishing characteristic functions of the
measurement errors. We derive upper bounds on the risk of the proposed
estimators and provide sufficient conditions under which zeros of the
corresponding characteristic function have no effect on estimation accuracy.
Moreover, we show that the derived conditions are also necessary in some
specific problem instances.
