For certain Fréchet spaces of entire functions of several variables satisfying some specified growth conditions, we define a constant coefficient differential operator as the transpose of a convolution operation in the dual space of continuous linear functionals and show that for in one of these spaces, their always exists a solution of the differential equation 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 comme la transposée d’une opération de convolution dans l’espace dual de fonctionnelles linéaires continues et nous montrons que pour dans un de ces espaces, il existe toujours une solution de l’équation différentielle 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}, pages = {211--238}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {22}, number = {1}, year = {1972}, doi = {10.5802/aif.404}, mrnumber = {48 #11552}, zbl = {0221.35005}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.404/} }
TY - JOUR AU - Gruman, Lawrence TI - The growth of entire solutions of differential equations of finite and infinite order JO - Annales de l'Institut Fourier PY - 1972 SP - 211 EP - 238 VL - 22 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.404/ DO - 10.5802/aif.404 LA - en ID - AIF_1972__22_1_211_0 ER -
%0 Journal Article %A Gruman, Lawrence %T The growth of entire solutions of differential equations of finite and infinite order %J Annales de l'Institut Fourier %D 1972 %P 211-238 %V 22 %N 1 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/aif.404/ %R 10.5802/aif.404 %G en %F 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/articles/10.5802/aif.404/
[1] The minimum modulus of small integral and subharmonic functions, Proc. London Math. Soc. (3) 12 (1962), 445-495. | MR | Zbl
,[2] Analytic Functions of Several Complex Variables, Englewood Cliffs, N.J., Prentice-Hall, (1965). | MR | Zbl
and ,[3] An Introduction to complex analysis in several variables, Princeton, N.J., Van Nostrand, 1966. | MR | Zbl
,[4] 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] Distribution of zeros of entire functions, Translations of Mathematical Monographs, Vol. 5, A.M.S., Providence, R.I. 1964. | Zbl
,[6] 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] 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] Equations différentielles d'ordre infini, Bull. Soc. math. France, 95, (1967), 109-154. | Numdam | Zbl
,[9] Linear Partial Differential Equations with Constant Coefficients, New York, Gordon and Breach (1966). | Zbl
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