## 3.2 Well-posedness of the deconvolution in S ∗ 

Well-posedness requires that a unique solution to the problem exist and that this solution be continuous in some ”reasonable topology” (Hadamard, 1923). Here most of the results consider the topology of generalized functions which is weaker than, say, the uniform or L$_{1}$ norm for corresponding subspaces, thus well-posedness may hold in this topology when it does not hold in the usual norms. But well-posedness may not always obtain even in this weak topol- ogy. An example is provided that involves deconvolution with a supersmooth distribution. 

Consider a convolution pair of spaces ( A, B ) in S $^{∗}$, denote by C the space of elements in S ∗ represented by convolutions of generalized functions from ( A, B ) . Well-posedness of the deconvolution would require that for the given f ∈ B and any sequence of w$_{n}$ ∈ C that converges to some w ∈ C in the topology of S ∗ : w$_{n}$ → w, the sequence of g$_{n}$ ∈ A such that g$_{n}$ ∗ f = w$_{n}$ would converge to g ∈ A, with g ∗ f = w. By Theorem 1 this convergence can be restated for the product pair of spaces (Γ , Φ) where the product pair is the image of the convolution pair ( A, B ) under Fourier transform; denote by Π ∈ S ∗ the space of products of elements from the pair (Γ , Φ). Then well- posedness can be restated in terms of the Fourier transforms: the problem is well-posed in S ∗ if for any ε$_{n}$ → ε where ε$_{n}$, ε ∈ Π the corresponding sequence γ$_{n}$ = φ − $^{1}$ε$_{n}$ converges to γ = φ − $^{1}$ε. 

If φ − 1 ∈ O$_{M}$ then by hypocontinuity of the product, for any sequence ε$_{n}$ − ε that converges to zero in S ∗ the corresponding sequence γ$_{n}$ − γ also 

