known classical spaces of generalized functions discussed in the books by Schwartz (1966) and Gel’fand and Shilov (1964); reference is also made to related spaces as presented e.g. by Sobolev (1992). Define a space of well behaved functions, sometimes called test functions; denote a generic such space by G . The functions in G are defined on the real space, R $^{d}$, but may take values in the complex space since we consider characteristic functions and Fourier transforms that can take complex values. Two widely used spaces of test functions are G = D and G = S. The space D is the linear topological space of infinitely differentiable functions each defined on a compact support, so that D ⊂ C $^{∞}$( R $^{d}$) , where C $^{∞}$( R $^{d}$) is the space of all infinitely differentiable functions; to converge in D the sequence of functions should be supported on a common bounded set and converge uniformly itself as well as have uniformly converging derivatives of all orders. To define the space S first introduce some notation. For any vector of non-negative integers m = ( m$_{1}$, ...m$_{d}$ ) and vector t ∈ R d denote by t m the product t m$_{1}$ 1 ...t m$_{d}$ d and by ∂ m the differentiation operator ∂ m 1 ∂x m 1 1 ... ∂ m d ∂x m d d . The space S ⊂ C $^{∞}$( R $^{d}$) of rapidly decreasing functions is then defined as: for any d − dimensional vectors of integers m, l, where l = (0 , ... 0) corresponds to the function itself, | t | is the vector of absolute values of vector t , t → ∞ S = { ψ ∈ C $^{∞}$( R $^{d}$) : ∣ t | $^{m}$| ∂ $^{l}$ψ ( t ) ∣ = o (1) as t → ∞ } , 13