Oka–Weil theorem explained

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

-convex subset of X, then every holomorphic function in an open neighborhood of K can be approximated uniformly on K by holomorphic functions on (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

• Oka coherence theorem

Bibliography

• Jorge. Mujica . The Oka–Weil theorem in locally convex spaces with the approximation property . Séminaire Paul Krée Tome 4. 1977–1978 . 1–7 . 0401.46024.
  • Oka . Kiyoshi . Sur les fonctions analytiques de plusieurs variables. II–Domaines d'holomorphie . Journal of Science of the Hiroshima University, Series A . 1937 . 7 . 115–130 . 10.32917/hmj/1558576819. free .
• Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes . Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris. 118–121. Remmert. Reinhold. 1956. 243. fr. 0070.30401.
• Weil . André . L'intégrale de Cauchy et les fonctions de plusieurs variables . Mathematische Annalen . 1935 . 111 . 178–182 . 10.1007/BF01472212. 120807854 .
• Book: 10.1007/978-1-4757-3878-0_7. The Oka—Weil Theorem . Banach Algebras and Several Complex Variables . Graduate Texts in Mathematics . 1976 . Wermer . John . 35 . 36–42 . 978-1-4757-3880-3 .

Further reading

• Kiyoshi. Oka. Sur les fonctions analytiques de plusieurs variables IV. Domaines d'holomorphie et domaines rationnellement convexes. Japanese Journal of Mathematics. 17. 1941. 517–521. 10.4099/jjm1924.17.0_517. free. – An example where Runge's theorem does not hold.
• 10.4153/CJM-2014-024-1. Global Holomorphic Functions in Several Noncommuting Variables . 2015 . Agler . Jim . McCarthy . John E. . Canadian Journal of Mathematics . 67 . 2 . 241–285 . 1305.1636 . 120834161 .

Notes and References

1. Book: J.E.. Fornaess . Forstneric . F . Wold . E.F . Daniel . Breaz . Michael Th. . Rassias . Advancements in Complex Analysis – Holomorphic Approximation . The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan . 2020 . . 133–192. 10.1007/978-3-030-40120-7. 1802.03924 . 978-3-030-40119-1 . 220266044 .