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lark
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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 \subset C^n 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 \notin D such that
|p(w)| \leq max(|p(z)|, z \in K), for all polynomials p.
Laura
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 \subset C^n 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 \notin D such that
|p(w)| \leq max(|p(z)|, z \in K), for all polynomials p.
Laura
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