coordinate-wise; thus the functions in S go to zero at infinity faster than any power as do their derivatives of any order. A sequence in S converges if in every bounded region each ∣ t | $^{l}$| ∂ $^{k}$ψ ( t ) ∣ converges uniformly. The generalized functions space G ∗ is the dual space, the space of linear continuous functionals on G. For any b ∈ G ∗ and any ψ ∈ G denote the value of the functional b applied to ψ by ( b, ψ ) . The topology is defined by weak convergence: a sequence b$_{n}$ ∈ G ∗ converges to b ∈ G ∗ if for any ψ ∈ G the sequence of the values of the functionals converges: ( b$_{n}$, ψ ) → ( b, ψ ) . Sobolev (1992) gives a general definition (in 1.8) where he points out a subtle distinction between the functional and a generalized function. Any generalized function, b ∈ G $^{∗}$, can be defined by an equivalence class { b$_{n}$ } of weakly converging sequences of test functions b$_{n}$ ∈ G : b = { { b$_{n}$ } : b$_{n}$ ∈ G, such that for any ψ ∈ G, lim n →∞ ∫ b$_{n}$ ( t ) ψ ( t ) dt = ( b, ψ ) < ∞ } , where ∫ · dt denotes the multivariate integral over R $^{d}$, over-bar indicates com- plex conjugate for complex-valued functions and ( b, ψ ) provides the value of the functional b ∈ G ∗ for ψ ∈ G. However, the same functional can be repre- sented by different generalized functions corresponding to different spaces G. For example, consider the δ − function. This is a linear continuous functional on the space C (0) of continuous functions as well as on D or S and provides ( δ, ψ ) = ψ (0); it can be represented as an equivalence class of δ − convergent sequences of continuous functions as well as of functions from D or S . This 14