Abstract:
Differential operators of infinite order with symbols that are infinitely differentiable functions (of Gevrey class) in some domain in $\mathbf R^n$ 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 $\mathbf C^n$.
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