## 5.1 Consistency of deconvolution: solving (15) 

The well-posedness condition directly applies to the measurement error de- convolution problem where φ is known and the conditions apply only to estimators of observables. The requirement of boundedness of the sequence of the estimators in the space S ∗ would follow if the norms of the functions were bounded in probability; L$_{p}$ are the spaces typically used in the litera- ture (e.g. Fan, 1991; Carrasco and Florens, 2010). But here more generally consistency of plug-in deconvolution holds for any probability distribution; the only important restriction is on φ − 1 to satisfy (13) ; that is for the mea- surement error not to be supersmooth. 

Moreover, the functions in (2) and the corresponding (15) need not be generalized densities and characteristic functions; the conditions for consis- tent estimation follow from those for well-posedness and are essentially that the two functions belong to a convolution pair (equivalently, Fourier trans- forms to a product pair) and that the known function be continuous and φ − 1 satisfy (13) . 

For example, consider in the univariate case a generalized density func- tion, w. One can use empirical distribution functions to estimate the corre- sponding distribution function, then the estimator of generalized density by generalized derivatives provides as an estimator the sum of delta-functions: ˆ w$_{n}$ = 1 n n ∑ j =1 δ ( x$_{j}$ ) , where ( δ ( x$_{j}$ ) , ψ ) = ψ ( x$_{j}$ ) . Then the corresponding Fourier transform ˆ ε$_{n}$ = F t ( ˆ w$_{n}$ ) is given by a continuous function ˆ ε$_{n}$ ( s ) = 1 n n ∑ j =1 e is $^{$^{T}$x$_{j}$}$. 

