Geodesic
Integral representation of entire functions and differential operators of infinite order
Izvestiya. Mathematics , Tome 59 (1995) no. 4, pp. 839-846

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In this article we derive an integral representation in certain spaces of entire functions of exponential type in $\mathbb C^n$. To this end we use the isomorphism, given by the Laplace operator, between these spaces and the corresponding spaces of ultradistributions. Using this integral representation these functions admit a well-defined action of differential operators of infinite order with specific conditions on the characteristic function. 
@article{IM2_1995_59_4_a8,
     author = {O. V. Odinokov},
     title = {Integral representation of entire functions and differential operators of infinite order},
     journal = {Izvestiya. Mathematics },
     pages = {839--846},
     publisher = {mathdoc},
     volume = {59},
     number = {4},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1995_59_4_a8/}
}
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O. V. Odinokov. Integral representation of entire functions and differential operators of infinite order. Izvestiya. Mathematics , Tome 59 (1995) no. 4, pp. 839-846. http://geodesic.mathdoc.fr/item/IM2_1995_59_4_a8/