are truthfully reported. Indeed if there is a mass at zero the error distribution is a mixture with the delta-function that has Fourier transform equal to 1, then φ is separated from zero and condition (13) for φ − 1 holds. A further extension is possible to deconvolution where the generalized function γ is not necessarily a continuous function and can be singular. Then additional differentiability conditions are needed to ensure that γ and φ are in a product pair. Let Γ = S ∗ $_{l,m}$( V ) with l = ( l$_{1}$, ..., l$_{d}$ ); φ be a bounded continuous function that has continuous derivatives of any order ≤ l$_{1}$ + ... + l$_{d}$ ; then a product ε = γφ exists in S ∗ for any γ ∈ Γ . Theorem 4 Assume γ, γ$_{n}$ ∈ S ∗ $_{l,m}$( V ) , and φ is a continuous regular function that has continuous derivatives ∂ l$_{i}$ for all subvectors l$_{i}$ of l ; ε = γφ, ε$_{n}$ = γ$_{n}$φ for n = 1 , 2 , ... ; and ε$_{n}$ − ε → 0 in S $^{∗}$. If φ − 1 and all ∂ l$_{i}$ ( φ − $^{1}$) for any subvector l$_{i}$ of each l satisfy (13) then ε$_{n}$φ − 1 exists in S ∗ and F t − $^{1}$( ε$_{n}$φ − $^{1}$) → g in S $^{∗}$. Proof. Denote by c$_{n}$ and c the continuous functions for which ε$_{n}$ and ε can be defined via the operator ∂ $^{l}$. Consider the functional ( φ − $^{1}$ε$_{n}$ − φ − $^{1}$ε, ψ ) = ( φ − $^{1}$∂ $^{l}$( c$_{n}$ − c ) , ψ ) ; it can be extended to the functional where κ ( l$_{1}$, l$_{2}$ ) is the corresponding integer coefficient arising from differen- tiation of the product. All the functionals ( c$_{n}$ − c, ∂ $^{l$_{1}$}$( φ − $^{1}$) ∂ $^{l$_{2}$}$ψ ) that enter into the linear combinations on the right-hand side satisfy the conditions of ( c$_{n}$ − c, ( − 1) | l | ∂ $^{l}$( φ − $^{1}$ψ ) ) = ( − 1) | l | ∑ ( l$_{1}$ ; l$_{2}$ )= l κ ( l$_{1}$, l$_{2}$ ) ( c$_{n}$ − c, ∂ $^{l$_{1}$}$( φ − $^{1}$) ∂ $^{l$_{2}$}$ψ ) , 38