that Assumption 2 holds. Define the subset of all continuously differentiable characteristic functions by ¯ C $^{(1)}$. In model 5 equations involve a product of the regression function g with f$_{x}$ ∗ . Products of generalized functions in S ∗ do not always exist and so addi- tional restrictions are needed in that model. If g is an arbitrary element of S $^{∗}$, then for the product to exist, f$_{x}$ ∗ should be in O$_{M}$ . On the other hand, if f$_{x}$ ∗ is an arbitrary generalized density it is sufficient that g and h$_{k}$ belong to the space of d times continuously differentiable functions with derivatives that are majorized by polynomial functions for gf$_{x}$ ∗ , h$_{k}$f$_{x}$ ∗ to be elements of S $^{∗}$. Indeed, the value of the functional h$_{k}$f$_{x}$ ∗ for an arbitrary ψ ∈ S is defined by here F is the distribution (ordinary bounded) function and this integral ex- ists because ψ and all its derivatives go to zero at infinity faster than any polynomial function. Denote by ¯ S B, 1 the space of continuously differentiable functions g ∈ S ∗ such that the functions h$_{k}$ ( x ) = x$_{k}$g ( x ) are also continu- ously differentiable with all derivatives majorized by polynomial functions . Since the products are in S ∗ then the Fourier transforms of the products are defined in S $^{∗}$. Further restrictions requiring the Fourier transforms of the products gf$_{x}$ ∗ and h$_{k}$f$_{x}$ ∗ to be continuously differentiable functions in S ∗ would remove any restrictions on f$_{u}$ for the convolution to exist. Denote the space of all continuously differentiable functions in S ∗ by ¯ S $^{(1)}$. ( h$_{k}$f$_{x}$ ∗ , ψ ) = ( − 1) d ∫ F$_{x}$ ∗ ( x ) ∂ (1 ,..., $^{1)}$( h$_{k}$ ( x ) ψ ( x )) dx ; 33