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

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.

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.

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     title = {The growth of entire solutions of differential equations of finite and infinite order},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Institut Fourier},
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Gruman, Lawrence. The growth of entire solutions of differential equations of finite and infinite order. Annales de l'Institut Fourier, Tome 22 (1972) no. 1, pp. 211-238. doi : 10.5802/aif.404. http://archive.numdam.org/articles/10.5802/aif.404/

[1] P.D. Barry, The minimum modulus of small integral and subharmonic functions, Proc. London Math. Soc. (3) 12 (1962), 445-495. | MR | Zbl

[2] R.C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Englewood Cliffs, N.J., Prentice-Hall, (1965). | MR | Zbl

[3] L. Hormander, An Introduction to complex analysis in several variables, Princeton, N.J., Van Nostrand, 1966. | MR | Zbl

[4] P. Lelong, Non-continuous indicators for entire functions of n ≥ 2 variables and finite order, Proc. Sym. Pure Math. 11 (1968), p. 285-297. | MR | Zbl

[5] B.Ja. Levin, Distribution of zeros of entire functions, Translations of Mathematical Monographs, Vol. 5, A.M.S., Providence, R.I. 1964. | Zbl

[6] B. Malgrange, Existence et approximations des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier, Grenoble, t. 6, 1955-1956, 271-355 (Thèse Sc. math., Paris, 1955). | Numdam | MR | Zbl

[7] A. Martineau, Sur les fonctionnelles analytiques et la transformation de Fourier-Borel, J. Anal. math. Jérusalem, t. 11, (1963), 1-164 (Thèse Sc. math., Paris, 1963). | Zbl

[8] A. Martineau, Equations différentielles d'ordre infini, Bull. Soc. math. France, 95, (1967), 109-154. | Numdam | Zbl

[9] F. Treves, Linear Partial Differential Equations with Constant Coefficients, New York, Gordon and Breach (1966). | Zbl

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