Deconvolution using the deconvolution kernel (as e.g. in An and Hu, 2012) is applied when the density of the mismeasured variable is assumed to exist and belong in some L$_{p}$, the deconvolution kernel incorporates spectral cut-off by employing estimators of the form ˜ ε$_{n}$ = ˆ ε$_{n}$ ( s ) I ( | s | < T$_{n}$ ) , where T$_{n}$ goes to ∞ as n → ∞ at some rate; the indicator function allows for smoothing in the inverse Fourier transform. By the results here consistency in S ∗ is ensured for both estimators. In- deed since empirical distribution converges uniformly in probability to the distribution function, it thus converges in S $^{∗}$, then the generalized derivatives also converge in S ∗ and by continuity of the Fourier transform estimators ˆ ε$_{n}$ and for T$_{n}$ → ∞ also ˜ ε$_{n}$ converge to ε in probability. All these functions are continuous and bounded and only (13) for φ − 1 is needed for consistency of the deconvolution solution in S ∗ and the spectral cut-off is not required. The following remark summarizes this result. Remark 2. If the functions in equation (2) are all generalized density functions, the known characteristic function φ is such that φ − 1 satisfies (13) , { z$_{i}$ } n i =1 is a random sample from the distribution with the generalized density w, the deconvolution estimator is consistent in S ∗ : for any finite set ψ$_{1}$, ...ψ$_{v}$ ∈ S the random vector ((ˆ g$_{n}$ − g, ψ$_{1}$ ) , ..., (ˆ g$_{n}$ − g, ψ$_{v}$ )) T →$_{p}$ 0 . ˆ g$_{n}$ = F t − 1 ( φ − $^{1}$ˆ ε$_{n}$ ) where ˆ ε$_{n}$ ( s ) = 1 n n ∑ j =1 e is $^{T}$z$_{j}$ 51