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Polynomial convexity.

  1. Nov 25, 2008 #1
    I've been reading Gunning and Rossi's book on complex analysis in several variables (good book!).
    They define something called "polynomial convexity" for a domain D. "domain" = connected open set, I think.
    The point of polynomial convexity is that if D [tex]\subset C^n[/tex] is polynomially convex, then an analytic function on D can be approximated by polynomials, and the approximation is uniform on compact subsets of D.
    Is the converse true? i.e. if D isn't polynomially convex, is there a function that's analytic on D that can't be approximated by polynomials uniformly on compact subsets?
    If D isn't polynomially convex, that means there's a compact subset K of D and a point w [tex]\notin[/tex] D such that
    |p(w)| [tex]\leq[/tex] max(|p(z)|, z [tex]\in[/tex] K), for all polynomials p.
    Last edited: Nov 25, 2008
  2. jcsd
  3. Nov 27, 2008 #2
    I got an answer to this. The converse isn't true in [tex]C^n, n >1[/tex], but it's true in C.
    See this link
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