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[edit]

The Oka–Weil theorem states that if X is a Stein space and K is a compact ${\mathcal {O}}(X)$ -convex subset of X, then every holomorphic function in an open neighborhood of K can be approximated uniformly on K by holomorphic functions on ${\mathcal {O}}(X)$ (i.e. by polynomials).^[1]

Applications[edit]

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.

Oka coherence theorem

Jorge, Mujica (1977–1978). "The Oka–Weil theorem in locally convex spaces with the approximation property". Séminaire Paul Krée Tome 4: 1–7. 0401.46024.

Zbl

Noguchi, Junjiro (2019), (PDF), Kodai Math. J., 42 (3): 566–586, arXiv:1704.07726, doi:10.2996/kmj/1572487232, S2CID 119697608

"A Weak Coherence Theorem and Remarks to the Oka Theory"

Oka, Kiyoshi (1937). . Journal of Science of the Hiroshima University, Series A. 7: 115–130. doi:10.32917/hmj/1558576819.

"Sur les fonctions analytiques de plusieurs variables. II–Domaines d'holomorphie"

Remmert, Reinhold (1956). . Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris (in French). 243: 118–121. Zbl 0070.30401.

"Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes"

Weil, André (1935). "L'intégrale de Cauchy et les fonctions de plusieurs variables". Mathematische Annalen. 111: 178–182. :10.1007/BF01472212. S2CID 120807854.

doi

Wermer, John (1976). "The Oka—Weil Theorem". Banach Algebras and Several Complex Variables. Graduate Texts in Mathematics. Vol. 35. pp. 36–42. :10.1007/978-1-4757-3878-0_7. ISBN 978-1-4757-3880-3.

doi

Oka, Kiyoshi (1941). . Japanese Journal of Mathematics. 17: 517–521. doi:10.4099/jjm1924.17.0_517. – An example where Runge's theorem does not hold.

"Sur les fonctions analytiques de plusieurs variables IV. Domaines d'holomorphie et domaines rationnellement convexes"

Agler, Jim; McCarthy, John E. (2015). "Global Holomorphic Functions in Several Noncommuting Variables". Canadian Journal of Mathematics. 67 (2): 241–285. :1305.1636. doi:10.4153/CJM-2014-024-1. S2CID 120834161.