converges to zero in S $^{∗}$, thus γ$_{n}$ converges to γ in S ∗ and since the Fourier transform is continuously invertible in S $^{∗}$, g$_{n}$ = F t − $^{1}$( ε$_{1}$$_{n}$φ − 1 n ) → g in S $^{∗}$. (b) Modify the proof of ( a ) by stating existence of products based on the assumption in ( b ) rather than assuming the specific product pair ( S $^{∗}$, O$_{M}$ ) . Thus well-posedness obtains for the solution to the errors in variables regression with Berkson type instruments that provides (3) as long as φ − 1 and all the φ − 1 n are all in the same class Φ( m, V )); e.g. a set of the characteristic functions would satisfy this if they are not supersmooth and the growth of all φ − 1 n is bounded by the same order polynomial. With the condition (a) which does not restrict γ ∈ S ∗ the Theorem will hold if the supports of φ, φ$_{n}$ are bounded or if these were Fourier transforms of functions with some singularity points (e.g. from distributions with mass points), when φ would include a constant. The condition (b) would require imposing restrictions on degree of singularity of γ and further differentiability conditions on φ and φ$_{n}$ of the type imposed in Theorem 4 to ensure that all the products exist. Even if in the model φ is a characteristic function, φ$_{n}$ need not necessarily be characteristic functions, but just continuous functions that satisfy the conditions. This result did not require any restrictions on γ and thus on the regression function in the model beyond belonging to S $^{∗}$. Under Theorem 3(a) a similar result holds after imposing the conditions on γ, γ$_{n}$ rather than φ, φ$_{n}$ to ensure that generalized functions κ$_{kn}$ = ( γ$_{n}$ ) $^{′}$$_{k}$γ − 1 n belong to O$_{M}$ and to some Φ( ˜ m, V ) . 43