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 ${\displaystyle {\mathcal {O}}(X)}$-convex subset of X, then every holomorphic function in an open neighborhood of K can be approximated uniformly on K by holomorphic functions on ${\displaystyle {\mathcal {O}}(X)}$ (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

References[]

Bibliography[]

• Jorge, Mujica (1977–1978). "The Oka–Weil theorem in locally convex spaces with the approximation property". Séminaire Paul Krée Tome 4: 1–7. Zbl 0401.46024.
• Noguchi, Junjiro (2019), "A Weak Coherence Theorem and Remarks to the Oka Theory" (PDF), Kodai Math. J., 42 (3): 566–586, arXiv:1704.07726, doi:10.2996/kmj/1572487232
• 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.
• Weil, André (1935). "L'intégrale de Cauchy et les fonctions de plusieurs variables". Mathematische Annalen. 111: 178–182. doi:10.1007/BF01472212. S2CID 120807854.

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.

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