identification in some of the models. This point was made in e.g. Carrasco and Florens (2010), Evdokimov and White (2011) and is expressed here in the context of operating in S $^{∗}$. Support restriction for the solution may imply that only partial identification will be provided. However, even in partially identified models some features of interest (see, e.g. Matzkin, 2007) could be identified thus some questions could be addressed even in the absence of full identification. A common example of incomplete identification which nevertheless provides important information is Gaussian deconvolution of a blurred image of a car obtained from a traffic camera; the filtered image is still not very good, but the licence plate number is visible for forensics. 

Well-posedness conditions are emphasized here. The well-known defini- tion by Hadamard (1923) defines well-posedness via three conditions: exis- tence of a solution, uniqueness of the solution and continuity in some suitable topology. The first two are essentially identification. Since here we shall be defining the functions in subclasses of S ∗ we shall consider continuity in the topology of this generalized functions space. This topology is weaker than the topologies in functions spaces, such as the uniform or L$_{p}$ topologies; thus differentiating the distribution function to obtain a density is a well-posed problem in S $^{∗}$, by contrast, even in the class of absolutely continuous distri- butions with uniform metric where identification for density in the space L$_{1}$ holds, well-posedness however does not obtain (see discussion in Zinde-Walsh, 2011). But even though in the weaker topology of S ∗ well-posedness obtains more widely, for the problems considered here some additional restrictions 

