Complex Analysis
Cauchy–Fantappiè transformation and mutual dualities between A(Ω) and A(Ω˜) for lineally convex domains
[La transformation de Cauchy–Fantappiè et les dualités mutuelles entre A(Ω) et A(Ω˜) pour des domaines linéellement convexes]
Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1155-1158.

Dans cette Note nous présentons les résultats suivants : (i) la description par la transformation de Cauchy–Fantappiè des fonctionnelles analytiques, des dualités mutuelles entre le (DFS)-espace A(Ω) des fonctions holomorphes dans le domaine linéellement convexe Ω de Cn(n2) avec croissance polynomiale près de la frontière de ∂Ω, et le (FS)-espace A(Ω˜) des fonctions holomorphes dans lʼintérieur de lʼensemble conjugué Ω˜ qui sont dans C(Ω˜) ; (ii) lʼexistence dʼensembles suffisants dénombrables dans A(Ω) et A(Ω˜) ; (iii) la possibilité (respectivement, lʼimpossibilité) de représentation des fonctions de A(Ω˜) (respectivement, A(Ω)) sous la forme de séries de fractions partielles.

In this Note we present the following results: (i) a description, via the Cauchy–Fantappiè transformation of analytic functionals, of the mutual dualities between the (DFS)-space A(Ω) of holomorphic functions in a bounded lineally convex domain Ω of Cn(n2) with polynomial growth near the boundary ∂Ω, and the (FS)-space A(Ω˜) of holomorphic functions in the interior of the conjugate set Ω˜ that are in C(Ω˜); (ii) the existence of countable sufficient sets in A(Ω) and A(Ω˜); (iii) a possibility (respectively, the failure) of representing functions from A(Ω˜) (respectively, A(Ω)) in the form of series of partial fractions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.10.013
Abanin, A.V. 1, 2 ; Khoi, Le Hai 3

1 Southern Institute of Mathematics (SIM), Vladikavkaz 362027, The Russian Federation
2 Southern Federal University (SFU), Rostov-on-Don 344090, The Russian Federation
3 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University (NTU), 637371 Singapore
@article{CRMATH_2011__349_21-22_1155_0,
     author = {Abanin, A.V. and Khoi, Le Hai},
     title = {Cauchy{\textendash}Fantappi\`e transformation and mutual dualities between $ {A}^{-\infty }(\Omega )$ and $ {A}^{\infty }(\tilde{\Omega })$ for lineally convex domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1155--1158},
     publisher = {Elsevier},
     volume = {349},
     number = {21-22},
     year = {2011},
     doi = {10.1016/j.crma.2011.10.013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2011.10.013/}
}
TY  - JOUR
AU  - Abanin, A.V.
AU  - Khoi, Le Hai
TI  - Cauchy–Fantappiè transformation and mutual dualities between $ {A}^{-\infty }(\Omega )$ and $ {A}^{\infty }(\tilde{\Omega })$ for lineally convex domains
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1155
EP  - 1158
VL  - 349
IS  - 21-22
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2011.10.013/
DO  - 10.1016/j.crma.2011.10.013
LA  - en
ID  - CRMATH_2011__349_21-22_1155_0
ER  - 
%0 Journal Article
%A Abanin, A.V.
%A Khoi, Le Hai
%T Cauchy–Fantappiè transformation and mutual dualities between $ {A}^{-\infty }(\Omega )$ and $ {A}^{\infty }(\tilde{\Omega })$ for lineally convex domains
%J Comptes Rendus. Mathématique
%D 2011
%P 1155-1158
%V 349
%N 21-22
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2011.10.013/
%R 10.1016/j.crma.2011.10.013
%G en
%F CRMATH_2011__349_21-22_1155_0
Abanin, A.V.; Khoi, Le Hai. Cauchy–Fantappiè transformation and mutual dualities between $ {A}^{-\infty }(\Omega )$ and $ {A}^{\infty }(\tilde{\Omega })$ for lineally convex domains. Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1155-1158. doi : 10.1016/j.crma.2011.10.013. http://www.numdam.org/articles/10.1016/j.crma.2011.10.013/

[1] A.V. Abanin, Le Hai Khoi, Pre-dual of the function algebra A(D) and representation of functions in Dirichlet series, Complex Anal. Oper. Theory (, available online on Feb 18, 2010). | DOI

[2] Abanin, A.V.; Khoi, Le Hai Dual of the function algebra A(D) and representation of functions in Dirichlet series, Proc. Amer. Math. Soc., Volume 138 (2010), pp. 3623-3635

[3] Andersson, M.; Passare, M.; Sigurdsson, R. Complex Convexity and Analytic Functionals, Birkhäuser, 2004

[4] Barrett, D. Duality between A and A on domains with non-degenerate corners, Contemp. Math. AMS, Volume 185 (1995), pp. 77-87

[5] Bierstedt, K.D.; Meise, R. Summers, J. Math. Anal. Appl., Volume 277 (2003), pp. 651-669

[6] Bonet, J.; Domański, P. Sampling sets and sufficient sets for A, J. Math. Anal. Appl., Volume 277 (2003), pp. 651-669

[7] Choi, Y.J.; Khoi, Le Hai; Kim, K.T. On an explicit construction of weakly sufficient sets for the function algebra A, Compl. Var. Elliptic Equation, Volume 54 (2009), pp. 879-897

[8] Derzhavets, B.A. Spaces of functions, analytic in some linearly convex domains of Cn, having prescribed behavior near the boundary, Izv. Vyssh. Uchebn. Zaved. Mat., Volume 29 (1985) no. 6, pp. 10-13 (in Russian). English translation in Soviet Math. (Iz. VUZ), 29, 6, 1985, pp. 11-14

[9] Epifanov, O.V. Variations of weakly sufficient sets in spaces of analytic functions, Izv. Vyssh. Uchebn. Zaved. Mat., Volume 30 (1986) no. 7, pp. 50-56 (in Russian). English translation in Soviet Math. (Iz. VUZ), 30, 7, 1986, pp. 67-74

[10] Horowitz, C.A.; Korenblum, B.; Pinchuk, B. Sampling sequences for A, Michigan Math. J., Volume 44 (1997), pp. 389-398

[11] Khoi, Le Hai Sets of uniqueness, weakly sufficient sets and sampling sets for A(B), Bull. Korean Math. Soc., Volume 47 (2010), pp. 933-950

[12] Kiselman, C.O. A study of the Bergman projection in certain Hartogs domains, Proc. Symp. Pure Math., Volume 52 (1991), pp. 219-232

[13] Schneider, D.M. Sufficient sets for some spaces of entire functions, Trans. Amer. Math. Soc., Volume 197 (1974), pp. 161-180

Cité par Sources :