If g is an ordinary function that represents a regular element in S ∗ the infinite differentiability condition on f$_{x}$ ∗ can be relaxed to simply requiring continuous first derivatives. 

In models 6 and 7 if the generalized density function for the error, f$_{u}$, decreases faster than any polynomial (all moments need to exist for that), so that f$_{u}$ ∈ O ∗ $_{C}$, then g could be any generalized function in S $^{∗}$; this will of course hold if f$_{u}$ has bounded support. Generally, the more moments the error is assumed to have, the fewer restrictions on the regression function g are needed to satisfy the convolution equations of the model and the exchange formula. The models 6, 7 satisfy the assumptions for any error u when support of generalized function g is compact (as for the ”sum of peaks”), then g ∈ E ∗ ⊂ S $^{∗}$, where E ∗ is the space of generalized functions with compact support. More generally the functions g and all the h$_{k}$ could belong to the space O ∗ C of generalized functions that decrease at infinity faster than any polynomial, and still no restrictions need to be placed on u. 

Denote for any generalized density function f$_{·}$ the corresponding charac- teristic function, F t ( f$_{·}$ ) , by φ$_{·}$. Denote Fourier transform of the (generalized) regression function g, F t ( g ) , by γ. 

The following table summarizes some fairly general sufficient conditions on the models that place restrictions on the functions themselves or on the characteristic functions of distributions in the models that will ensure that Assumption 2 is satisfied and a solution exists. The nature of these assump- tions is to provide restrictions on some of the functions that allow the others 

