On the duality between $ {A}^{-\infty }(D)$ and $ {A}_{D}^{-\infty }$ for convex domains

Alexander V. Abanin; Le Hai Khoi

doi:10.1016/j.crma.2009.06.008

Complex Analysis

On the duality between

A^{- \infty} (D)

and

A_{D}^{- \infty}

for convex domains
[Sur la dualité entre

A^{- \infty} (D)

A_{D}^{- \infty}

pour des domaines convexes]

Présenté par : Jean-Pierre Demailly

Alexander V. Abanin ¹ ; Le Hai Khoi ²

¹ Southern Institute of Mathematics (SIM), Vladikavkaz 362027 and Southern Federal University (SFU), Rostov-on-Don 344090, The Russian Federation
² Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University (NTU), 637371 Singapore

Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 863-866.

Résumé
Abstract

Le but de cette Note est de démontrer que la transformation de Laplace des fonctionnelles analytiques établit une dualité mutuelle entre les espaces $A^{- \infty} (D)$ et $A_{D}^{- \infty}$ (D étant un domaine convexe borné dans $C^{N}$ ) et que des fonctions de $A_{D}^{- \infty}$ peuvent être représentées sous la forme de séries de Dirichlet avec fréquence de D.

The goal of this Note is to prove that the Laplace transformation of analytic functionals establishes the mutual duality between the spaces $A^{- \infty} (D)$ and $A_{D}^{- \infty}$ (D being a bounded convex domain in $C^{N}$ ) and that functions from $A_{D}^{- \infty}$ can be represented in a form of Dirichlet series with frequencies from D.

Reçu le : 2009-04-12
Accepté le : 2009-06-08
Publié le : 2009-07-02

DOI : 10.1016/j.crma.2009.06.008

Affiliations des auteurs :

Alexander V. Abanin ¹ ; Le Hai Khoi ²

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     author = {Alexander V. Abanin and Le Hai Khoi},
     title = {On the duality between $ {A}^{-\infty }(D)$ and $ {A}_{D}^{-\infty }$ for convex domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {863--866},
     publisher = {Elsevier},
     volume = {347},
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     year = {2009},
     doi = {10.1016/j.crma.2009.06.008},
     language = {en},
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Alexander V. Abanin; Le Hai Khoi. On the duality between $ {A}^{-\infty }(D)$ and $ {A}_{D}^{-\infty }$ for convex domains. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 863-866. doi : 10.1016/j.crma.2009.06.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.06.008/

Bibliographie
Cité par

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[3] Y.J. Choi, L.H. Khoi, K.T. Kim, On an explicit construction of weakly sufficient sets for the function algebra $A^{- \infty} (Ω)$ , Compl. Variables & Elliptic Equations, in press

[4] L. Hörmander The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer, 1983

[5] L.H. Khoi Espaces conjugués, ensembles faiblement suffisants discrets et systèmes de représentation exponentielle, Bull. Sci. Math. (2), Volume 113 (1989), pp. 309-347

[6] Yu.F. Korobeinik Inductive and projective topologies. Sufficient sets and representing systems, Math. USSR-Izv., Volume 28 (1987), pp. 529-554

[7] A. Martineau Equations différentielles d'ordre infini, Bull. Soc. Math. France, Volume 95 (1967), pp. 109-154

[8] S.N. Melikhov (DFS)-spaces of holomorphic functions invariant under differentiation, J. Math. Anal. Appl., Volume 297 (2004), pp. 577-586

[9] E.J. Straube Harmonic and analytic functions admitting a distribution boundary value, Ann. Scuola Norm. Sup. Pisa, Volume 11 (1984), pp. 559-591

[10] H. Whitney Analytic extension of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., Volume 36 (1934), pp. 63-89

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