polynomial convexity
Graven Water
variables (good book!).
They define something called "polynomial convexity" for a domain D in
n-dimensional complex space. "domain" = connected open set, I think.
The point of polynomial convexity is that if D 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 not in D such that
|p(w)| <= max(|p(z)|, z in K), for all polynomials p.
Laura
David C. Ullrich
wrote:
>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 in
>n-dimensional complex space. "domain" = connected open set, I think.
>
>The point of polynomial convexity is that if D is polynomially convex,
>then an analytic function on D can be approximated by polynomials,
>and the approximation is uniform on compact subsets of D.
I don't know if I'd say that's the _point_ to polynomial
convexity...
>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?
No (at least not in more than one variable). Say D is the set of all
z in C^2 with 1/2 < |z| < 1. Any holomorphic function on D actually
extends to a function holomorphic in the unit ball (Hartogs), and
hence can be approximated on compact sets by polynomials.
But D is certainly not polynomially convex.
>If D isn't polynomially convex, that means there's a compact subset K
>of D and a point w not in D such that
>|p(w)| <= max(|p(z)|, z in K), for all polynomials p.
>
>Laura
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Graven Water
complex plane, though, it's true.
Suppose K is the compact subset of D whose polynomial convex hull is not
contained in D. Then C-K has a bounded component B, that is not contained
in D. Take a point w in B-D. Then boundary(B) is contained in K, and a
compact set L can be defined by taking the union of closed balls that are
centered at the points of boundary(B) and contained in D.
f(z) = 1/(z-w) is analytic on D, but by the maximum modulus theorem it can't
be uniformly approximated on L by polynomials.
Laura
David C. Ullrich
wrote:
>Thanks for the counterexample. It seems like if D is a region in the
>complex plane, though, it's true.
Yes. In one variable the notion of "polynomially convex" doesn't
usually come up because (at least for connected open sets) it's
the same as "simply connected".
One of many reasons why "several complex variables" is
not usually taken to include the case of one complex variable...
>Suppose K is the compact subset of D whose polynomial convex hull is not
>contained in D. Then C-K has a bounded component B, that is not contained
>in D. Take a point w in B-D. Then boundary(B) is contained in K, and a
>compact set L can be defined by taking the union of closed balls that are
>centered at the points of boundary(B) and contained in D.
"Defined" is a little strong here - the union of _all_ such balls is
not compact. But yes, there is a compact set L that consists of
_a_ union of such balls (including one centered at every point
of the boundary of B).
>f(z) = 1/(z-w) is analytic on D, but by the maximum modulus theorem it can't
>be uniformly approximated on L by polynomials.
>Laura
David C. Ullrich
Graven Water
> "Defined" is a little strong here - the union of _all_ such balls is
> not compact. But yes, there is a compact set L that consists of
> _a_ union of such balls (including one centered at every point
> of the boundary of B).
Well yes - I thought of flapping my fingers to the effect that
"there's a nonzero
minimum distance between boundary(B) and C-D, so take the union of
closed balls centered on the points of boundary(B) with radius half
that distance, and that is compact", but didn't.
There is an obvious follow-on question here: Suppose the
domain D in n-dimensional complex space is maximal in the sense that
D is not properly contained in another domain D', such that all
functions holomorphic on D can be extended to D'. *Then* if D is
not polynomially convex, is there a function holomorphic on D that
can't be approximated by polynomials uniformly on compact sets.
But then Gunning and Rossi are about to tell me all about domains of
holomorphy and maybe, whether the polynomials are dense in the
ring of analytic functions on a domain of holomorphy. I'll see :)
Laura