no. 4
Dualité des espaces de fonctions entières en dimension infinie
Annales de l'Institut Fourier, Tome 26 (1976) no. 4, pp. 151-195.

On étudie ici quelques espaces de fonctions holomorphes dans des domaines localement convexes, ayant comme cas particuliers les espaces de Fock holomorphes. Les espaces duaux sont caractérisés avec la transformation de Fourier-Borel pour des types d’holomorphie appropriés. On montre que ces espaces de fonctions sont de Fréchet-Schwartz (resp. de Silva, resp. nucléaires) quand leurs domaines sont des espaces de Silva (resp. de Fréchet-Schwartz, resp. nucléaires). Les conditions de croissance p-sommable des séries de Taylor qui y interviennent sont équivalentes aux conditions de croissance exponentielle employées par Martineau dans l’étude des équations différentielles d’ordre infini et sont plus maniables que celles-ci dans des questions de dualité en dimension infinie.

Dual pairs of spaces of holomorphic functions on locally convex domains are studied, yielding the holomorphic Fock spaces as particular cases. The duality is with respect to the Fourier-Borel transformation for appropriate holomorphy types. These functions spaces are shown to be Fréchet-Schwartz spaces (resp. Silva spaces, resp. nuclear). The p-summable growth conditions of Taylor series that intervene in their definitions are equivalent to the exponential growth conditions employed by Martineau in the study of differential equations of infinite order, and are more manageable in the treatment of duality questions in infinite dimension.

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     author = {Dwyer III, Thomas A. W.},
     title = {Dualit\'e des espaces de fonctions enti\`eres en dimension infinie},
     journal = {Annales de l'Institut Fourier},
     pages = {151--195},
     publisher = {Imprimerie Durand},
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     volume = {26},
     number = {4},
     year = {1976},
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     language = {fr},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.636/}
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Dwyer III, Thomas A. W. Dualité des espaces de fonctions entières en dimension infinie. Annales de l'Institut Fourier, Tome 26 (1976) no. 4, pp. 151-195. doi : 10.5802/aif.636. https://aif.centre-mersenne.org/articles/10.5802/aif.636/

[1]R. Aron, Holomorphy types for open subsets of a Banach space, Studia Math., 45 (à paraître). | EuDML | MR | Zbl

[2]R. Aron, Holomorphic functions on balanced subsets of a Banach space, Bull. Amer. Math. Soc., 78 (1972), 624-627. | MR | Zbl

[3]R. Aron, Tensor products of holomorphic functions, Nederl. Akad. Wetensch. Indag. Math., 35 (76) (1973), Fasc. 3, 192-202. | MR | Zbl

[4]V. Bargmann, Remarks on a Hilbert space of analytic functions, Proc. Nat. Acad. Sci., 48 (1962), 199-204. | MR | Zbl

[5]P. Berezin, The method of second quantization, Pure and Applied Physics, Vol. 24, Academic Press; New York-London, 1966. | MR | Zbl

[6]Ph. Boland, Some spaces of entire and nuclearly entire functions on a Banach space, J. Reine Angew. Math., 270 (1974), 38-60. | EuDML | MR | Zbl

[7]Ph. Boland, Espaces pondérés de fonctions entières et de fonctions entières nucléaires sur un espace de Banach, C.R. Acad. Sci., Paris, Sér. A-B, 275 (1972), A587-A590. | MR | Zbl

[8]Ph. Boland, Malgrange theorem for entire functions on nuclear spaces, Proc. on infinite-dimensional holomorphy, Lecture Notes in Mathematics N° 364, Springer-Verlag, Berlin-Heidelberg-New York, 1974. | MR | Zbl

[9]Ph. Boland, Holomorphic mappings on DFN (dual of Fréchet nuclear) spaces, Séminaire P. Lelong (Analyse) 14e année, 1973-1974. | Zbl

[10]S. Dineen, Holomorphy types on a Banach space, Studia Math., 39 (1971), 241-288. | MR | Zbl

[11]S. Dineen, Holomorphically significant properties of topological vector spaces, Proc. Colloque international du C.N.R.S. sur űLes fonctions analytiques de plusieurs variablesƇ, Paris, 14-20 Juin 1972 (à paraître). | Zbl

[12]S. Dineen, Holomorphic functions and surjective limits, Proc. on infinite dimensional holomorphy, Lecture Notes in Mathematics N° 364, Springer-Verlag, Berlin-Heidelberg-New York, 1974. | MR | Zbl

[13]T. Dwyer, Partial differential equations in Fischer-Fock spaces for the Hilbert-Schmidt holomorphy type, Bull. Amer. Math. Soc., 77 (1971), 725-730. MR 44 # 7288. | MR | Zbl

[14]T. Dwyer, Holomorphic Fock representations and partial differential equations on countably Hilbert spaces, Bull. Amer. Soc., 79 (1973), 1045-1050. | MR | Zbl

[15]T. Dwyer, Partial differential equations in holomorphic Fock spaces, Functional analysis and applications, Lecture Notes in Mathematics N° 384, Springer-Verlag, Berlin-Heidelberg-New York, 1974. | MR | Zbl

[16]T. Dwyer, Holomorphic representation of tempered distributions and weighted Fock spaces, Analyse fonctionnelle et applications, Hermann, Paris, 1975, pp 95-118. | MR | Zbl

[17]T. Dwyer, Convolution equations for vector-valued entire functions of bounded nuclear type, Trans. Amer. Math. Soc. (à paraître). | Zbl

[18]T. Dwyer, Dualité des espaces de fonctions entières en dimension infinie, C.R. Acad. Sci., Paris, Sér. A 280 (1975), 1439-1442. | MR | Zbl

[19]T. Dwyer, Équations différentielles d'ordre infini dans des espaces localement convexes, C.R. Acad. Sci., Paris, Sér. A 281 (1975), 163-166. | MR | Zbl

[20]K. Floret et J. Wloka, Einführung in die Theorie der Lokalkonvexen] Räume, Lecture Notes in Mathematics N° 56, Springer-Verlag, Berlin-Heidelberg-New York, 1968. | MR | Zbl

[21]I. Gelfand et N. Vilenkin, Generalized functions, Vol. 4, Academic Press, New York-London, 1964.

[22]Ch. Gupta, Malgrange theorem for nuclearly entire functions of bounded type on a Banach space, Notas de Matematica N° 37, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1968. | MR | Zbl

[23]Ch. Gupta, Convolution operators and holomorphic mappings on a Banach space, Séminaire d'Analyse Moderne N° 2, Univ. de Sherbrooke Québec, 1969. | Zbl

[24]Ch. Gupta, On the Malgrange theorem for nuclearly entire functions of bounded type on a Banach space, Nederl. Akad. Wetensch, Proc., Ser. A 73 = Indag. Math., 32 (1970), 356-358. | MR | Zbl

[25]A. Martineau, Sur les fonctionnelles analytiques et la transformation de Fourier-Borel, Jour. d'Anal. Math., XI (1963), 1-164. | MR | Zbl

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

[27] M. De Matos, Sur le théorème d'approximation et d'existence de Malgrange-Gupta, C.R. Acad. Sci., Paris, Sér. A-B 271 (1970), A1258-A-1259. MR 44 # 3105. | Zbl

[28] M. De Matos, Holomorphic mappings and domains of holomorphy, Monografias do Centro Brasileiro de Pesquisas Fisicas, Vol. 27, 1970. | Zbl

[29]L. Nachbin, Topology on spaces of holomorphic mappings, Ergebn sse der Mathematik und ihrer Grenzgebiete, Band 47, Springer-Verlag, Berlin-Heidelberg-New York, 1969. MR 40 # 7787. | MR | Zbl

[30]L. Nachbin, Concerning holomorphy types for Banach spaces, Proc. Internat. Colloq. on Nuclear Spaces and Ideals in Operator Algebras, Studia Math., 48 (1970), 407-412. MR 43 # 3787. | MR | Zbl

[31]L. Nachbin, Convolution operators in spaces of nuclearly entire functions on a Banach space, Functional analysis and related fields, Springer-Verlag, Berlin-Heidelberg-New York, 1970. | Zbl

[32]L. Nachbin, Convoluções em funções inteiras nucleares, Atas da 2a Quinzena de Análise Funcional e Equações Diferenciais Parciais, Sociedade Brasileira de Matemática, 1972.

[33]Yu. Novozhilov et A. Tulub, The method of functionals in the quantum theory of fields, Uspehi. Fiz. Nauk, 61 (1957), 53-102 = Russian Tracts on advanced Math and Math. Phys., Vol. 5, Gordon and Breach, New York, 1961. MR 25 # 959.

[34]A. Robertson et W. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Mathematical Physics N° 53, Cambridge University Press, 1964. | MR | Zbl

[35]J. Rzewuski, On Hilbert space of entire functionals, Bull. Acad. Polon. Sci., Sér. Math. Astronom. Phys., 17 (1969), 453-458 ; 459-466 ; 571-578. MR 40 # 6243 ; 6244 ; 6245. | MR | Zbl

[36]J. Rzewuski, Hilbert of functional power series, Reports on Math. Phys., 1 (1970/1971), N° 3, 195-210. MR 44 # 1349. | MR | Zbl

[37]B. Taylor, Some locally convex spaces of entire functions, Proc. Symp. Pure Math., Vol. 11, Entire functions and related parts of analysis, Amer. Math. Soc., Providence, R.I., 1968. | Zbl

[38]F. Trèves, Topological vector spaces, distributions and kernels, Pure and Applied Mathematics, Vol. 25, Academic Press, New York-London, 1967. | MR | Zbl

[39]F. Trèves, Linear partial differential equations with constant coefficients : existence, approximation and regularity of solutions, Math. and its applications, Vol. 6, Gordon and Breach, New York, 1966. | MR | Zbl

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