Associated and quasi associated homogeneous distributions (generalized functions)

In this paper analysis of the concept of associated homogeneous distributions (generalized functions) is given, and some problems related to these distributions are solved. It is proved that (in the one-dimensional case) there exist only associated homogeneous distributions of order k=1. We introduce a definition of quasi associated homogeneous distributions which is a natural generalization of the notion of associated homogeneous distributions and provide a mathematical description of all quasi associated homogeneous distributions and their Fourier transforms. It is proved that the class of quasi associated homogeneous distributions coincides with the class of distributions introduced in [I.M. Gel'fand, G.E. Shilov, Generalized Functions, vol. 1, Properties and Operations, Academic Press, New York, 1964, Chapter I, §4] as the class of associated homogeneous distributions. For the multidimensional case it is proved that f is a quasi associated homogeneous distribution if and only if it satisfies an Euler type system of differential equations. A new type of Γ-functions generated by quasi associated homogeneous distributions is defined.