Uniform approximation on manifolds

• 成果类型: Article
• 署名作者: Izzo, Alexander J.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.1.2
• 发表日期: 2011
• 页码: 55-73
• 关键词: complex manifold; holomorphic approximation; polynomial-approximation; c-n; ALGEBRAS; submanifolds; invariant; THEOREMS

摘要

It is shown that if A is a uniform algebra generated by a family (13 of complex-valued C-1 functions on a compact C-1 manifold-with-boundary M, the maximal ideal space of A is M, and E is the set of points where the differentials of the functions in Phi fail to span the complexified cotangent space to M, then A contains every continuous function on M that vanishes on E. This answers a 45-year-old question of Michael Freeman who proved the special case in which the manifold M is two-dimensional. More general forms of the theorem are also established. The results presented strengthen results due to several mathematicians.