function classes that are usually considered, but allows for some useful gen- eralizations. The next subsection provides the necessary definitions and some of the implications of working in the space S $^{∗}$. 2.1 The space of generalized functions S $^{∗}$. The space S ∗ is the dual space, i.e. the space of continuous linear function- als on the space S of functions. The theory of generalized functions is in Schwartz (1966); relevant details are summarized in Zinde-Walsh (2012). In this subsection the main definitions and properties are reproduced. Recall the definition of S. 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}$1 1 ... ∂ m d ∂x m d d ; C$_{∞}$ is the space of infinitely differentiable (real or complex- valued) functions on R $^{d}$. The space S ⊂ C$_{∞}$ of test functions is defined as: for any k = ( k$_{1}$, ...k$_{d}$ ) , l = ( l$_{1}$, ...l$_{d}$ ) , where k = (0 , ... 0) corresponds to the function itself, t → ∞ coordinate-wise; thus the functions in S go to zero at infinity faster than any power as do their derivatives; they are rapidly decreasing functions. A sequence in S converges if in every bounded region each ∣ t $^{l}$∂$^{k}$ψ ( t ) ∣ converges uniformly. Then in the dual space S ∗ any b ∈ S ∗ represents a linear functional on 9 S = { ψ ∈ C$_{∞}$ ( R $^{d}$) : | t $^{l}$∂$^{k}$ψ ( t ) | = o (1) as t → ∞ } ,