product pairs of subspaces of their Fourier transforms. The classical pairs of spaces (Schwartz, 1966) are the convolution pair ( S $^{∗}$, O∗ $_{C}$) and the corresponding product pair ( S $^{∗}$, O$_{M}$ ) , where O ∗ C is the subspace of S ∗ that contains rapidly decreasing (faster than any polynomial) generalized functions and O$_{M}$ is the space of infinitely differentiable functions with every derivative growing no faster than a polynomial at infinity. These pairs are important in that no restriction is placed on one of the generalized functions that could be any element of space S $^{∗}$; the other belongs to a space that needs to be correspondingly restricted. A disadvantage of the classical pairs is that the restriction is fairly severe, for example, the requirement that a characteristic function be in O$_{M}$ implies existence of all moments for the random variable. Relaxing this restriction would require placing constraints on the other space in the pair; Zinde-Walsh (2012) introduces some pairs that incorporate such trade-offs. In some models the product of a function with a component of the vector of arguments is involved,such as d ( x ) = x$_{k}$a ( x ) , then for Fourier transforms Ft ( d ) ( s ) = − i ∂ ∂s $_{k}$Ft ( a )( s ); the multiplication by a variable is transformed into ( − i ) times the corresponding partial derivative. Since the differentia- tion operators are continuous in S ∗ this transformation does not present a problem. Assumption 2. The functions a ∈ A, b ∈ B, are such that ( A, B ) form a convolution pair in S $^{∗}$. Equivalently, Ft ( a ) , Ft ( b ) are in the corresponding product pair of spaces. 31