  Truncating the Fourier transform averaged by means of a generalized Hausdorff
operator, we approximate the adjoint to that Hausdorff operator of the given
function. We find the formulas for the rate of approximation in various metrics
in terms of the parameter of truncation and the components of the Hausdorff
operator. Explicit rates of approximation and comparison with approximate
identities are given in the case of Lipschitz $\alpha$ continuous functions.
