I've been reading Gunning and Rossi's book on complex analysis in several variables (good book!).(adsbygoogle = window.adsbygoogle || []).push({});

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.

Laura

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# Polynomial convexity.

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