The growth of entire solutions of differential equations of finite and infinite order

Lawrence Gruman

Annales de l'institut Fourier (1972)

  • Volume: 22, Issue: 1, page 211-238
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Gruman, Lawrence. "The growth of entire solutions of differential equations of finite and infinite order." Annales de l'institut Fourier 22.1 (1972): 211-238. <http://eudml.org/doc/74066>.

@article{Gruman1972,
abstract = {For certain Fréchet spaces of entire functions of several variables satisfying some specified growth conditions, we define a constant coefficient differential operator $\check\{\alpha \}$ 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 $\check\{\alpha \}(x) = f$ in the same space.},
author = {Gruman, Lawrence},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {211-238},
publisher = {Association des Annales de l'Institut Fourier},
title = {The growth of entire solutions of differential equations of finite and infinite order},
url = {http://eudml.org/doc/74066},
volume = {22},
year = {1972},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 22
IS - 1
SP - 211
EP - 238
AB - For certain Fréchet spaces of entire functions of several variables satisfying some specified growth conditions, we define a constant coefficient differential operator $\check{\alpha }$ 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 $\check{\alpha }(x) = f$ in the same space.
LA - eng
UR - http://eudml.org/doc/74066
ER -

References

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  1. [1] P.D. BARRY, The minimum modulus of small integral and subharmonic functions, Proc. London Math. Soc. (3) 12 (1962), 445-495. Zbl0196.41603MR25 #3172
  2. [2] R.C. GUNNING and H. ROSSI, Analytic Functions of Several Complex Variables, Englewood Cliffs, N.J., Prentice-Hall, (1965). Zbl0141.08601MR31 #4927
  3. [3] L. HORMANDER, An Introduction to complex analysis in several variables, Princeton, N.J., Van Nostrand, 1966. Zbl0138.06203MR34 #2933
  4. [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. Zbl0177.34102MR38 #3464
  5. [5] B.Ja. LEVIN, Distribution of zeros of entire functions, Translations of Mathematical Monographs, Vol. 5, A.M.S., Providence, R.I. 1964. Zbl0152.06703
  6. [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). Zbl0071.09002MR19,280a
  7. [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). Zbl0124.31804
  8. [8] A. MARTINEAU, Equations différentielles d'ordre infini, Bull. Soc. math. France, 95, (1967), 109-154. Zbl0167.44202
  9. [9] F. TREVES, Linear Partial Differential Equations with Constant Coefficients, New York, Gordon and Breach (1966). Zbl0164.40602

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