Regularization with suitable spectral cut-off in the case of supersmooth er- ror distributions typically provides a sequence of estimators that will converge in norm, albeit slowly for supersmooth distributions (Fan, 1991); convergence to the limit in the normed space implies convergence in the topology of S $^{∗}$. However, if φ is not supersmooth the class of generalized functions where consistent estimation holds does not include all probability distributions. In- deed spectral cut-off estimation provides generalized densities that are a limit of inverse Fourier transforms of truncated generalized functions. Schwartz (1964, pp.271-273) gives a characterization of any function in S ∗ with Fourier transform that has compact support (in a cube | z$_{k}$ | < C, k = 1 , ...d ) based on Wiener-Paley theorem. Such a function is a continuous function g that can be extended to a entire analytic function G of a complex argument and is of exponential type ≤ 2 C , meaning Thus as long as g is such a function or a limit of such functions it can be expressed via the regularized solution. As in Schwartz the subspace of all functions of exponential type (for any finite C) can also be considered. However, if g does not belong to this subspace regularized solutions may not converge to it even in S $^{∗}$. lim | z |→∞ log | G ( z ) | | z$_{1}$ | + ... + | z$_{d}$ | ≤ 2 C. 52