Oka–Weil theorem

The Oka–Weil theorem states that if X is a Stein space and K is a compact $\backslash mathcal\{O\}(X)$-convex subset of X, then every holomorphic function in an open neighborhood of K can be approximated uniformly on K by holomorphic functions on $\backslash mathcal\{O\}(X)$ (in particular, by polynomials).{{cite book |first1=J.E.|last1 =Fornaess |last2=Forstneric |first2=F |last3=Wold |first3=E.F |editor1-first=Daniel |editor1-last=Breaz |editor2-first=Michael Th. |editor2-last=Rassias |title=Advancements in Complex Analysis – Holomorphic Approximation |chapter=The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan |date=2020 |publisher=Springer Nature |pages=133–192|doi=10.1007/978-3-030-40120-7|arxiv=1802.03924 |isbn =978-3-030-40119-1 |s2cid =220266044 }}

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

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• {{cite journal |last1=Jorge|first1=Mujica |title=The Oka–Weil theorem in locally convex spaces with the approximation property |journal=Séminaire Paul Krée Tome 4|year=1977–1978 |page=1–7 |zbl =0401.46024}}
• {{Citation | last1=Noguchi | first1=Junjiro | title=A Weak Coherence Theorem and Remarks to the Oka Theory |journal=Kodai Math. J.|volume=42|url=https://www.ms.u-tokyo.ac.jp/~noguchi/WeakcohOka_3.pdf | year=2019 | issue=3 | arxiv = 1704.07726|doi =10.2996/kmj/1572487232|pages=566–586| s2cid=119697608 }}
• {{cite journal |last1=Oka |first1=Kiyoshi |title=Sur les fonctions analytiques de plusieurs variables. II–Domaines d'holomorphie |journal=Journal of Science of the Hiroshima University, Series A |date=1937 |volume=7 |pages=115–130 |doi=10.32917/hmj/1558576819|doi-access=free }}
• {{cite journal |title =Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes |journal= Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris| pages=118–121| last1=Remmert|first1=Reinhold|year=1956|volume=243|url=https://gallica.bnf.fr/ark:/12148/bpt6k3195v/f118.item|lang=fr|zbl=0070.30401}}
• {{cite journal |last1=Weil |first1=André |title=L'intégrale de Cauchy et les fonctions de plusieurs variables |journal=Mathematische Annalen |date=1935 |volume=111 |pages=178–182 |doi=10.1007/BF01472212|s2cid=120807854 }}
• {{cite book |doi=10.1007/978-1-4757-3878-0_7|chapter=The Oka—Weil Theorem |title=Banach Algebras and Several Complex Variables |series=Graduate Texts in Mathematics |year=1976 |last1=Wermer |first1=John |volume=35 |pages=36–42 |isbn=978-1-4757-3880-3 }}

• {{Cite journal|first=Kiyoshi|last=Oka|title=Sur les fonctions analytiques de plusieurs variables IV. Domaines d'holomorphie et domaines rationnellement convexes|journal=Japanese Journal of Mathematics|volume=17|year=1941|pages=517–521|doi=10.4099/jjm1924.17.0_517|doi-access=free}} – An example where Runge's theorem does not hold.
• {{cite journal |doi=10.4153/CJM-2014-024-1| title=Global Holomorphic Functions in Several Noncommuting Variables | year=2015 | last1=Agler | first1=Jim | last2=McCarthy | first2=John E. | journal=Canadian Journal of Mathematics | volume=67 | issue=2 | pages=241–285 | arxiv=1305.1636 | s2cid=120834161 }}

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Category:Several complex variables

Category:Theorems in complex analysis

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