Approximation of holomorphic functions of infinitely many variables II

László Lempert

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 2, page 423-442
  • ISSN: 0373-0956

Abstract

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Let X be a Banach space and B ( R ) ⊂ X the ball of radius R centered at 0 . Can any holomorphic function on B ( R ) be approximated by entire functions, uniformly on smaller balls B ( r ) ? We answer this question in the affirmative for a large class of Banach spaces.

How to cite

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Lempert, László. "Approximation of holomorphic functions of infinitely many variables II." Annales de l'institut Fourier 50.2 (2000): 423-442. <http://eudml.org/doc/75424>.

@article{Lempert2000,
abstract = {Let $X$ be a Banach space and $B(R)\subset X$ the ball of radius $R$ centered at $0$. Can any holomorphic function on $B(R)$ be approximated by entire functions, uniformly on smaller balls $B(r)$? We answer this question in the affirmative for a large class of Banach spaces.},
author = {Lempert, László},
journal = {Annales de l'institut Fourier},
keywords = {holomorphic functions; Banach space; pseudoconvex domain; approximation problem; unconditional basis; Fréchet space; sheaf of germs},
language = {eng},
number = {2},
pages = {423-442},
publisher = {Association des Annales de l'Institut Fourier},
title = {Approximation of holomorphic functions of infinitely many variables II},
url = {http://eudml.org/doc/75424},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Lempert, László
TI - Approximation of holomorphic functions of infinitely many variables II
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 2
SP - 423
EP - 442
AB - Let $X$ be a Banach space and $B(R)\subset X$ the ball of radius $R$ centered at $0$. Can any holomorphic function on $B(R)$ be approximated by entire functions, uniformly on smaller balls $B(r)$? We answer this question in the affirmative for a large class of Banach spaces.
LA - eng
KW - holomorphic functions; Banach space; pseudoconvex domain; approximation problem; unconditional basis; Fréchet space; sheaf of germs
UR - http://eudml.org/doc/75424
ER -

References

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  1. [D1] S. DINEEN, Cousin's first problem on certain locally convex topological vector spaces, An. Acad. Brasil. Cienc., 48 (1976), 11-12. Zbl0354.46029MR55 #11290
  2. [D2] S. DINEEN, Complex Analysis in Locally Convex Spaces, North Holland, Amsterdam, 1981. Zbl0484.46044MR84b:46050
  3. [D3] S. DINEEN, Complex Analysis on Infinite Dimensional Spaces, Springer, Berlin, 1999. Zbl1034.46504MR2001a:46043
  4. [DS] N. DUNFORD, T. SCHWARTZ, Linear Operators I, John Wiley & Sons, New York, 1988. 
  5. [L1] L. LEMPERT, Approximation de fonctions holomorphes d'un nombre infini de variables, Ann. Inst. Fourier, 49-4 (1999), 1293-1304. Zbl0944.46046MR2001d:32027
  6. [L2] L. LEMPERT, The Dolbeault complex in infinite dimensions, II, J. Amer. Math. Soc., 12 (1999), 775-793. Zbl0926.32048MR2000e:32053
  7. [L3] L. LEMPERT, The Dolbeault complex in infinite dimensions III, manuscript.. Zbl0926.32048
  8. [M] P. MAZET, Analytic Sets in Locally Convex Spaces, North Holland, Amsterdam, 1984. Zbl0588.46032MR86i:32012
  9. [MV] R. MEISE and D. VOGT, Counterexamples in holomorphic functions on nuclear Fréchet spaces, Math. Z., 182 (1983), 167-177. Zbl0509.46041MR84m:46048
  10. [N] P. NOVERRAZ, Pseudo-convexité polynomiale et domaines d'holomorphie en dimension infinie, North Holland, Amsterdam, 1973. Zbl0251.46049MR50 #10814
  11. [P] I. PATYI, On the ∂-equation in a Banach space, Bull. Soc. Math. France, to appear. Zbl0967.32036
  12. [R] R. A. RYAN, Holomorphic mappings in l1, Trans. Amer. Soc., 302 (1987), 797-811. Zbl0637.46045MR88h:46089
  13. [S] I. SINGER, Bases in Banach spaces I-II, Springer, Berlin, 1981. Zbl0467.46020MR82k:46024

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