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Table 3) ensures existence of solutions to the convolution equations for models 1-7; this does not exclude multiple solutions and the next section provides a discussion of solutions for equations in Table 4. 

## 3.2 Classes of solutions; support and multiplicity of solutions 

Typically, support assumptions are required to restrict multiplicity of solu- tions; here we examine the dependence of solutions on the support of the functions. The results here also give conditions under which some zeros, e.g. in the characteristic functions, are allowed. Thus in common with e.g. Car- rasco and Florens (2010), Evdokimov and White (2011), distributions such as the uniform or triangular for which the characteristic function has isolated zeros are not excluded. The difference here is the extension of the considera- tion of the solutions to S ∗ and to models such as the regression model where this approach to relaxing support assumptions was not previously considered. 

Recall that for a continuous function ψ ( x ) on R d support is defined as the set W =supp( ψ ) , such that 

Support of a continuous function is an open set. 

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Generalized functions are functionals on the space S and support of a 

