Geodesic
Differential operators of infinite order with symbols of Gevrey class
Izvestiya. Mathematics , Tome 39 (1992) no. 2, pp. 1085-1096

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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 $\mathbf C^n$. 
@article{IM2_1992_39_2_a6,
     author = {O. V. Odinokov},
     title = {Differential operators of infinite order with symbols of {Gevrey} class},
     journal = {Izvestiya. Mathematics },
     pages = {1085--1096},
     publisher = {mathdoc},
     volume = {39},
     number = {2},
     year = {1992},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1992_39_2_a6/}
}
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O. V. Odinokov. Differential operators of infinite order with symbols of Gevrey class. Izvestiya. Mathematics , Tome 39 (1992) no. 2, pp. 1085-1096. http://geodesic.mathdoc.fr/item/IM2_1992_39_2_a6/