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be sought: any functions from these classes that enter into the convolution provide a result in S $^{∗}$. 

No assumptions are needed for existence of convolution and full generality of identification conditions in models 1,2 where the model assumptions im- ply that the functions represent generalized densities. For the other models including regression models convolution is not always defined in S $^{∗}$. Zinde- Walsh (2012) defines the concept of convolution pairs of classes of functions in S ∗ where convolution can be applied. 

To solve the convolution equations a Fourier transform is usually em- ployed, so that e.g. one transforms generalized density functions into charac- teristic functions. Fourier transform is an isomorphism of the space S $^{∗}$. The Fourier transform of a generalized function a ∈ S $^{∗}$, F t ( a ) , is defined as fol- lows. For any ψ ∈ S, as usual F t ( ψ )( s ) = ∫ ψ ( x ) e $^{isx}$dx ; then the functional F t ( a ) is defined by 

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The advantage of applying Fourier transform is that integral convolution equations transform into algebraic equations when the ”exchange formula” applies: 

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In the space of generalized functions S $^{∗}$, the Fourier transform and inverse Fourier transform always exist. As shown in Zinde-Walsh (2012) there is a di- chotomy between convolution pairs of subspaces in S ∗ and the corresponding 

