regression functions for which solutions can be obtained. In various applica- tions the object of interest may be represented by ”sum of peaks” function, such as a sum of delta functions see, e.g. Klann, Kuhn, Lorenz, Maass and Thiele (2007) where applications in astrophysics and mass spectroscopy are discussed; functions with sparse support or support that includes isolated points can arise in various applications. If g represents a regression func- tion, applying Fourier transform to the convolution equations in spaces of functions may require severe restrictions on the function. For example, in spaces of integrable functions (such as L$_{1}$, L$_{2}$ ) linear and polynomial regres- sion functions as well as distribution functions in binary choice models would be excluded. Again, a natural extension is to consider spaces of generalized functions where Fourier transforms are defined for functions that can grow at polynomial rates as well as for objects with sparse support. Spaces of generalized functions were utilized by Klann et al (2007) for sum of peaks re- gression, by Zinde-Walsh (2008) for generalized density functions; by Schen- nach (2007) and Zinde-Walsh (2009) for the problem in errors in variables univariate regression model with possible polynomial growth in the function. 

This paper examines convolution equations in generalized function spaces. The interest here focuses on the equation (2) where only the function g is unknown (deconvolution) and the system of equations (3) with two unknown functions. The generalized functions spaces considered here are described in detail in Schwartz (1966) and Gel’fand and Shilov (1964). 

Much of the paper is devoted to a theoretical development of the problem 

