be extended from the space D ( W$_{ν}$ ) to a zero functional on the space of continuous functions on W$_{ν}$, C $^{(0)}$( W$_{ν}$ ) ⊂ C $^{(0)}$( R $^{d}$) . Then (( γ − ˜ γ ) φ, ψ ) = 0 for any ψ ∈ C $^{(0)}$( W$_{ν}$ ) , but since φ is invertible as a continuous function on W$_{v}$ any continuous function ψ in C $^{(0)}$( W$_{ν}$ ) has the representation φ ˜ ψ for some ˜ ψ ∈ C $^{(0)}$( W$_{ν}$ ) . This implies that ( γ − ˜ γ, ψ ) is defined (equals zero) for any ψ thus the functional γ − ˜ γ extends to C $^{(0)}$( W$_{ν}$ ) as a zero functional. This is only possible if γ − ˜ γ is a zero functional in D $^{∗}$( W$_{ν}$ ) . Then γ − ˜ γ is a zero generalized function in D $^{∗}$( W ) . ■ 

Recall that the difference between D ∗ and S ∗ is in the ”tail behavior” only; on bounded sets the two spaces coincide. The following theorem provides deconvolution in S $^{∗}$. 

Theorem 2 Under Assumptions 1 and 2 assume that φ = F t ( f ) is a known continuous function; then for any W where supp( φ ) ⊃ W, the Fourier trans- form of g, γ, is uniquely defined on W ; if it is further known that supp( γ ) = W, then g is uniquely defined in S $^{∗}$. Uniqueness holds automatically if supp( φ ) = R $^{d}$. 

Proof. By Theorem 1, (15) holds in S $^{∗}$. Since supp( φ ) ⊃ W by Lemma 1 a unique solution γ to (15) exists in D $^{∗}$( W ). Since γ is the Fourier transform of g ∈ S $^{∗}$, γ ∈ S ∗ and defines an element in S $^{∗}$( W ) ⊂ D $^{∗}$( W ) uniquely. If support of γ is restricted to W a priori, then the solution is unique, thus when supp( φ ) = R $^{d}$, the generalized function γ is defined uniquely in S $^{∗}$. An inverse Fourier transform exists in S ∗ for any γ . It is then possible to recover 

