in D $^{∗}$( S $^{∗}$) . Schennach (2007) considered univariate errors in variables model with instrumental variables and a regression function, g, bounded by polyno- mials; these regression functions are in S $^{∗}$, convolution with any generalized density function from O ∗ C exists in S $^{∗}$. For regression functions g in subspaces of S ∗ where growth is more restricted, convolution with less rapidly declining f may exist. 

For a generalized function g denote by h$_{k}$ the product of g by the k th component of x ∈ R $^{d}$, x$_{k}$ , for a regular function g ( x ) this is h$_{k}$ ( x ) = x$_{k}$g ( x ). In many cases when the convolution g ∗ f is defined, the convolution h$_{k}$ ∗ f is also defined. Indeed, this is so in all the examples above: if g has compact support, so does any h$_{k}$ ; if g ∈ C $^{∞}$, it is true of h$_{k}$ as well, etc. Thus some models accommodate not only (2), but also other equations, e.g. providing (3). 

Assumption 3. The statistical model is such that in addition to Assump- tion 2, h$_{k}$ ∈ A , k = 1 , ..., d. 

## 2.3 Fourier transforms, exchange formula and some special convolution pairs 

Next, consider the Fourier transforms and the exchange formula. 

For the generalized functions in equations (3) denote Fourier transforms as γ = F t ( g ) , φ = F t ( f ) and ε$_{i}$ = F t ( w$_{i}$ ) . Recall that the Fourier transform always exists in S ∗ and is a continuous isomorphism. 

