Oka–Weil theorem
In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil.
Statement
The Oka–Weil theorem states that if X is a Stein space and K is a compact [math]\displaystyle{ \mathcal{O}(X) }[/math]-convex subset of X, then every holomorphic function in an open neighborhood of K can be approximated uniformly on K by holomorphic functions on [math]\displaystyle{ \mathcal{O}(X) }[/math] (i.e. by polynomials).[1]
Applications
Since Runge's theorem may not hold for several complex variables, the Oka–Weil theorem is often used as an approximation theorem for several complex variables. The Behnke–Stein theorem was originally proved using the Oka–Weil theorem.
See also
References
- ↑ Fornaess, J.E.; Forstneric, F; Wold, E.F (2020). "The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan". in Breaz, Daniel; Rassias, Michael Th.. Advancements in Complex Analysis – Holomorphic Approximation. Springer Nature. pp. 133–192. doi:10.1007/978-3-030-40120-7. ISBN 978-3-030-40119-1.
Bibliography
- Jorge, Mujica (1977–1978). "The Oka–Weil theorem in locally convex spaces with the approximation property". Séminaire Paul Krée Tome 4: 1–7.
- Noguchi, Junjiro (2019), "A Weak Coherence Theorem and Remarks to the Oka Theory", Kodai Math. J. 42 (3): 566–586, doi:10.2996/kmj/1572487232, https://www.ms.u-tokyo.ac.jp/~noguchi/WeakcohOka_3.pdf
- Oka, Kiyoshi (1937). "Sur les fonctions analytiques de plusieurs variables. II–Domaines d'holomorphie". Journal of Science of the Hiroshima University, Series A 7: 115–130. doi:10.32917/hmj/1558576819.
- Remmert, Reinhold (1956). "Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris 243: 118–121. https://gallica.bnf.fr/ark:/12148/bpt6k3195v/f118.item.
- Weil, André (1935). "L'intégrale de Cauchy et les fonctions de plusieurs variables". Mathematische Annalen 111: 178–182. doi:10.1007/BF01472212.
- Wermer, John (1976). "The Oka—Weil Theorem". Banach Algebras and Several Complex Variables. Graduate Texts in Mathematics. 35. pp. 36–42. doi:10.1007/978-1-4757-3878-0_7. ISBN 978-1-4757-3880-3.
Further reading
- Oka, Kiyoshi (1941). "Sur les fonctions analytiques de plusieurs variables IV. Domaines d'holomorphie et domaines rationnellement convexes". Japanese Journal of Mathematics 17: 517–521. doi:10.4099/jjm1924.17.0_517. – An example where Runge's theorem does not hold.
- Agler, Jim; McCarthy, John E. (2015). "Global Holomorphic Functions in Several Noncommuting Variables". Canadian Journal of Mathematics 67 (2): 241–285. doi:10.4153/CJM-2014-024-1.
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