Although the convolution operator is well known, this paper provides ex- plicitly convolution equations for a wide list of models for the first time. In many cases the analysis in the literature takes Fourier transforms as the start- ing point, e.g. characteristic functions for distributions of random vectors (as in the famous Kotlyarski lemma, 1967). The emphasis here on convolution equations for the models provides the opportunity to explicitly state non- parametric classes of functions defined by the model for which such equations hold, in particular, for densities, conditional densities and regression func- tions. The statistical model may give rise to different systems of convolution equations and may be over-identified in terms of convolution equations; some choices may be better suited to different situations, for example, here in Sec- tion 2 two sets of convolution equations (4 and 4a in Table 1) are provided for the same classical measurement error model with two measurements; it turns out that one of those allows to relax some independence conditions, while the other makes it possible to relax a support assumption in identification. Many of the convolution equations derived here are based on density-weighted con- ditional averages of the observables. 

The main distinguishing feature is that here all the functions defined by the model are considered within the space of generalized functions S $^{∗}$, the space of so-called tempered distributions (they will be referred to as generalized functions). This is the dual space, the space of linear continuous functionals, on the space S of well-behaved functions: the functions in S are infinitely differentiable and all the derivatives go to zero at infinity faster 

