Solvability of convolution equations in \(K_ M'\) (Q1113057)

Let S be a convolution operator in the space of distributions \(K_ M'\) [\textit{I. M. Gel'fand} and \textit{G. E. Shilov}, Generalized functions, Vol. 3 (1967; Zbl 0133.375)]. Then a necessary and sufficient condition on S to have \(S*K_ M'=K_ M^{*'}\) is given. The main result in this paper gives the necessay and sufficient conditions for solvability of determined systems of convolution equations in \(K_ M'\). Three theorems are proved in this context. The author remarks that the main theorem of the paper [\textit{S. Sznajder} and \textit{Z. Zielezny}, Pac. J. Math. 3, No.2, 539-544 (1976; Zbl 0334.46043)] is extended by the first theorem of the above three theorems. Lastly the author feels that the question of finding the necessary and sufficient conditions for solvability of undetermined systems of convolution equations in \(K_ M'\) is yet to be solved.