Uniqueness classes for the solution of Goursat's problem

Uniqueness classes for the solution of Goursat's problem, which consists in giving Cauchy initial conditions for each of the variables $t_i$, $ i=1,\dots,n$, are studied for linear partial differential equations with constant coefficients with two groups of variables: time $t=(t_1,\dots,t_n)$ and space $x=(x_1,\dots,x_m)$. The results obtained generalize a well-known theorem of Gel'fand and Shilov on uniqueness classes for the solution of Cauchy's problem.