DIFFERENTIAL-OPERATORS OF INFINITE-ORDER WITH SYMBOLS OF GEVREY CLASS

Differential operators of infinite order with symbols that are infinitely differentiable functions (of Gevrey class) in some domain in R(n) are considered. With the help of such operators a generalized Fourier transform of infinitely differentiable functions is constructed. For these operators a criterion for the of solvability of the Cauchy problem in some subclasses of exponential functions is proved. The results are similar to those of Dubinskii [1] for differential operators of infinite order with symbols analytic in some Runge domain in C(n).