The growth of entire solutions of differential equations of finite and infinite order
Annales de l'Institut Fourier, Volume 22 (1972) no. 1, p. 211-238

For certain Fréchet spaces of entire functions of several variables satisfying some specified growth conditions, we define a constant coefficient differential operator α ˇ as the transpose of a convolution operation in the dual space of continuous linear functionals and show that for f(z) in one of these spaces, their always exists a solution of the differential equation α ˇ(x)=f in the same space.

Pour certains espaces de Fréchet de fonctions entières de plusieurs variables qui satisfont à des conditions de croissance spécifiées, nous définissons un opérateur différentiel à coefficients constants α ˇ comme la transposée d’une opération de convolution dans l’espace dual de fonctionnelles linéaires continues et nous montrons que pour f(z) dans un de ces espaces, il existe toujours une solution de l’équation différentielle α ˇ(x)=f dans le même espace.

@article{AIF_1972__22_1_211_0,
     author = {Gruman, Lawrence},
     title = {The growth of entire solutions of differential equations of finite and infinite order},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {22},
     number = {1},
     year = {1972},
     pages = {211-238},
     doi = {10.5802/aif.404},
     zbl = {0221.35005},
     mrnumber = {48 \#11552},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1972__22_1_211_0}
}
Gruman, Lawrence. The growth of entire solutions of differential equations of finite and infinite order. Annales de l'Institut Fourier, Volume 22 (1972) no. 1, pp. 211-238. doi : 10.5802/aif.404. http://www.numdam.org/item/AIF_1972__22_1_211_0/

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