Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Division by holomorphic functions and convolution equations in infinite dimension


Authors: J.-F. Colombeau, R. Gay and B. Perrot
Journal: Trans. Amer. Math. Soc. 264 (1981), 381-391
MSC: Primary 46F25; Secondary 32A15, 46G20
MathSciNet review: 603769
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ E$ be a complex complete dual nuclear locally convex space (i.e. its strong dual is nuclear), $ \Omega $ a connected open set in $ E$ and $ \mathcal{E}(\Omega )$ the space of the $ {C^\infty }$ functions on $ \Omega $ (in the real sense). Then we show that any element of $ \mathcal{E}'(\Omega )$ may be divided by any nonzero holomorphic function on $ \Omega $ with the quotient as an element of $ \mathcal{E}'(\Omega )$. This result has for standard consequence a new proof of the surjectivity of any nonzero convolution operator on the space $ \operatorname{Exp} (E')$ of entire functions of exponential type on the dual $ E'$ of $ E$. As an application of the above division result and of a result of $ {C^\infty }$ solvability of the $ \overline \partial $ equation in strong duals of nuclear Fréchet spaces we study the solutions of the homogeneous convolution equations in $ \operatorname{Exp} (E')$ in terms of the zero set of their characteristic functions.


References [Enhancements On Off] (What's this?)

  • [1] V. I. Averbuh and O. G. Smoljanov, Different definitions of derivative in linear topological spaces, Uspehi Mat. Nauk 23 (1968), no. 4 (142), 67–116 (Russian). MR 0246118
  • [2] Philip J. Boland, Malgrange theorem for entire functions on nuclear spaces, Proceedings on Infinite Dimensional Holomorphy (Internat. Conf., Univ. Kentucky, Lexington, Ky., 1973) Springer, Berlin, 1974, pp. 135–144. Lecture Notes in Math., Vol. 364. MR 0420271
  • [3] -, Holomorphic functions on nuclear spaces, Publicaciones del Departamento de Analisis Mat., Univ. de Santiago de Compostela, (B) no. 16 (1976).
  • [4] J. F. Colombeau, $ {C^\infty }$ mappings in infinitely many dimensions and applications (preprint).
  • [5] J.-F. Colombeau, Sur les applications différentiables et analytiques au sens de J. Sebastiâo e Silva, Portugal. Math. 36 (1977), no. 2, 103–118 (1980) (French). MR 577416
  • [6] Jean-François Colombeau and Reinhold Meise, 𝐶^{∞}-functions on locally convex and on bornological vector spaces, Functional analysis, holomorphy, and approximation theory (Proc. Sem., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1978) Lecture Notes in Math., vol. 843, Springer, Berlin, 1981, pp. 195–216. MR 610831
  • [7] J.-F. Colombeau, R. Meise, and B. Perrot, A density result in spaces of Silva holomorphic mappings, Pacific J. Math. 84 (1979), no. 1, 35–42. MR 559625
  • [8] J.-F. Colombeau and B. Perrot, The Fourier-Borel transform in infinitely many dimensions and applications, Functional analysis, holomorphy, and approximation theory (Proc. Sem., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1978) Lecture Notes in Math., vol. 843, Springer, Berlin-New York, 1981, pp. 163–186. MR 610829
  • [9] J.-F. Colombeau and B. Perrot, Convolution equations in spaces of infinite-dimensional entire functions of exponential and related types, Trans. Amer. Math. Soc. 258 (1980), no. 1, 191–198. MR 554328, 10.1090/S0002-9947-1980-0554328-7
  • [10] -, The $ \overline \partial $ equation in DFN spaces, J. Math. Anal. Appl. (in press).
  • [11] Thomas A. W. Dwyer III, Differential operators of infinite order in locally convex spaces. I, Rend. Mat. (6) 10 (1977), no. 1, 149–179 (English, with Italian summary). MR 0482181
  • [12] Thomas A. W. Dwyer III, Differential operators of infinite order in locally convex spaces. II, Rend. Mat. (6) 10 (1977), no. 2-3, 273–293. MR 0482182
  • [13] Roger Gay, Sur un problème de division des fonctionnelles analytiques. Application aux fonctions moyenne-périodiques, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 11, Aii, A835–A838. MR 0427677
  • [14] I. M. Guelfand and G. E. Chilov, Les distributions, Traduit par G. Rideau. Collection Universitaire de Mathématiques, VIII, Dunod, Paris, 1962 (French). MR 0132390
  • [15] Henri Hogbe-Nlend, Bornologies and functional analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Introductory course on the theory of duality topology-bornology and its use in functional analysis; Translated from the French by V. B. Moscatelli; North-Holland Mathematics Studies, Vol. 26; Notas de Matemática, No. 62. [Notes on Mathematics, No. 62]. MR 0500064
  • [16] Lars Hörmander, On the division of distributions by polynomials, Ark. Mat. 3 (1958), 555–568. MR 0124734
  • [17] S. Łojasiewicz, Sur le problème de la division, Studia Math. 18 (1959), 87–136 (French). MR 0107168
  • [18] M. Z. Nashed, Differentiability and related properties of nonlinear operators: Some aspects of the role of differentials in nonlinear functional analysis, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 103–309. MR 0276840
  • [19] Albrecht Pietsch, Nuclear locally convex spaces, Springer-Verlag, New York-Heidelberg, 1972. Translated from the second German edition by William H. Ruckle; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 66. MR 0350360
  • [20] Jean-Pierre Ramis, Sous-ensembles analytiques d’une variété banachique complexe, Springer-Verlag, Berlin-New York, 1970 (French). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 53. MR 0293126
  • [21] Laurent Schwartz, Division par une fonction holomorphe sur une variété analytique complexe, Summa Brasil. Math. 3 (1955), 181–209 (1955) (French). MR 0139937
  • [22] Lucien Waelbroeck, Some theorems about bounded structures, J. Functional Analysis 1 (1967), 392–408. MR 0220040
  • [23] Lucien Waelbroeck, Topological vector spaces and algebras, Lecture Notes in Mathematics, Vol. 230, Springer-Verlag, Berlin-New York, 1971. MR 0467234

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46F25, 32A15, 46G20

Retrieve articles in all journals with MSC: 46F25, 32A15, 46G20


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0603769-9
Keywords: Infinite dimensional holomorphy, infinite dimensional distribution, convolution equation, $ \overline \partial $ equation, Fourier-Borel transform
Article copyright: © Copyright 1981 American Mathematical Society