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 Post subject: smooth vs analyticPosted: Sat, 17 Sep 2011 04:12:57 UTC
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Suppose is smooth with . Is necessary?

Suppose is analytic with . Is necessary?

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 Post subject: Re: smooth vs analyticPosted: Sat, 17 Sep 2011 10:00:38 UTC
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outermeasure wrote:
Suppose is smooth with . Is necessary?

Suppose is analytic with . Is necessary?

The first one:

Spoiler:
No. Hit any element of (assumed to have no real zeros for the denominator) with a partition of unity on a bounded interval, so that it vanishes outside.

Second one:

Spoiler:
Lemma: If is necessary, then is necessary.

Proof: It is clear that if is necessary then really is necessary, since any finite pole of a rational function would break our assumption that is analytic on all of . It is easy to see from here that if this is the case, then really --just test the polynomial on rational points and solve the resulting system of linear equations over the vector space

I'm going to think more on this tomorrow, it's 4:18 am here and I just pulled these two out, but I think the second one is going to require me to remember a couple more facts.

Great problem outermeasure! I like how the cases are so starkly different.

EDIT:
Spoiler:
realized I meant , not

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 Post subject: Re: smooth vs analyticPosted: Sat, 17 Sep 2011 10:18:51 UTC
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outermeasure wrote:
Suppose is smooth with . Is necessary?

Suppose is analytic with . Is necessary?

The first one:

Spoiler:
No. Hit any element of (assumed to have no real zeros for the denominator) with a bump function to make it zero outside some compact interval

Why would a bump function sends to ?

Spoiler:
In addition to using bump functions, you want to enumerate and use diagonal argument.

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 Post subject: Re: smooth vs analyticPosted: Sat, 17 Sep 2011 10:21:44 UTC
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outermeasure wrote:
outermeasure wrote:
Suppose is smooth with . Is necessary?

Suppose is analytic with . Is necessary?

The first one:

Spoiler:
No. Hit any element of (assumed to have no real zeros for the denominator) with a bump function to make it zero outside some compact interval

Why would a bump function sends to ?

Spoiler:
In addition to using bump functions, you want to enumerate and use diagonal argument.

Oh wow, I didn't even see you had responded, I actually modified my earlier argument, I might still need to do what you said, but I don't think so at a glance.

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 Post subject: Re: smooth vs analyticPosted: Sun, 18 Sep 2011 06:48:38 UTC
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Second one:

Spoiler:
Lemma: If is necessary, then is necessary.

Proof: It is clear that if is necessary then really is necessary, since any finite pole of a rational function would break our assumption that is analytic on all of . It is easy to see from here that if this is the case, then really --just test the polynomial on rational points and solve the resulting system of linear equations over the vector space

I'm going to think more on this tomorrow, it's 4:18 am here and I just pulled these two out, but I think the second one is going to require me to remember a couple more facts.

Great problem outermeasure! I like how the cases are so starkly different.

Yes, that's why I wrote instead of or .

Actually the cases are not that much different!

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 Post subject: Re: smooth vs analyticPosted: Sun, 18 Sep 2011 07:01:05 UTC
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outermeasure wrote:
Second one:

Spoiler:
Lemma: If is necessary, then is necessary.

Proof: It is clear that if is necessary then really is necessary, since any finite pole of a rational function would break our assumption that is analytic on all of . It is easy to see from here that if this is the case, then really --just test the polynomial on rational points and solve the resulting system of linear equations over the vector space

I'm going to think more on this tomorrow, it's 4:18 am here and I just pulled these two out, but I think the second one is going to require me to remember a couple more facts.

Great problem outermeasure! I like how the cases are so starkly different.

Yes, that's why I wrote instead of or .

Actually the cases are not that much different!

Really now? I know at least I cannot pull the same trick in the analytic context because of Liouville, but perhaps you mean there's an approach which handles both simultaneously which is not mine?

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 Post subject: Re: smooth vs analyticPosted: Sun, 18 Sep 2011 07:30:23 UTC
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Really now? I know at least I cannot pull the same trick in the analytic context because of Liouville, but perhaps you mean there's an approach which handles both simultaneously which is not mine?

Yes, there is an approach without using bump functions.

Spoiler:
Enumerating or is still crucial.

BTW, yes, there exists a bump function such that (but not every bump function has this property).

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 Post subject: Re: smooth vs analyticPosted: Mon, 19 Sep 2011 05:32:05 UTC
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Constructing a bump function with (gist):

Spoiler:
Enumerate the rationals . Take a bump function . If , define , else: choose a small interval with , , and a small such that , . Let be a bump function centred at , vanishing outside , with derivatives suitably bounded, and let . Inductively construct , making sure we avoid touching those previously done and that is small enough to guarantee has sufficiently small -norm () and . Finally, let .

Here is the common approach:

Spoiler:
Let be an enumeration of (or --- I'll just drop the mention unless it is significantly different). Recall vanishes at , and this serves as our building block.

Start with , . There are now two variants to how we choose the sequence --- either we use a cardinality argument, or we construct a power series directly.

Cardinality argument: We choose sufficiently small so that converges uniformly on compacts, e.g., let be a sequence of positive reals such that and . Then the limit is analytic and for all (since only finitely many summands are nonzero at ). Now the choice of sequence bijects with the continuum, but there are only countably many polynomials in (or for rational case).

Of course, you can dress that up in a diagonal argument instead --- choose your sufficiently small and yet disagrees with , where is an enumeration of (or ).

Constructing power series directly: Note that , so if we let (say) we have the coefficients of various 's will not interfere with each other, so it remains to choose the coefficients . Now choose with size sufficiently rapidly decreasing so the converges on compacts (and for the rational case, it is easy to make it transcendence --- indeed in this case this is guaranteed by the arbitrarily long string of zeros).

There is nothing special about or in this problem, except they are countable subrings with an element inside the punctured unit disc, so we can replace them with, e.g. or the algebraic closure of .

Edit: fix subscripts and typo.

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 Last edited by outermeasure on Mon, 19 Sep 2011 08:01:22 UTC, edited 1 time in total. fix subscripts and typo

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 Post subject: Re: smooth vs analyticPosted: Mon, 19 Sep 2011 05:37:53 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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outermeasure wrote:
Constructing a bump function with (gist):

Spoiler:
Enumerate the rationals . Take a bump function . If , define , else: choose a small interval with , , and a small such that , . Let be a bump function centred at , vanishing outside , with derivatives suitably bounded, and let . Inductively construct , making sure we avoid touching those previously done and that is small enough to guarantee has sufficiently small -norm () and . Finally, let .

Here is the common approach:

Spoiler:
Let be an enumeration of (or --- I'll just drop the mention unless it is significantly different). Recall vanishes at , and this serves as our building block.

Start with , . There are now two variants to how we choose the sequence --- either we use a cardinality argument, or we construct a power series directly.

Cardinality argument: We choose sufficiently small so that converges uniformly on compacts, e.g., let be a sequence of positive reals such that and . Then the limit is analytic and for all (since only finitely many summands are nonzero at ). Now the choice of sequence bijects with the continuum, but there are only countably many polynomials in (or for rational case).

Of course, you can dress that up in a diagonal argument instead --- choose your sufficiently small and yet disagrees with , where is an enumeration of (or ).

Constructing power series directly: Note that , so if we let (say) we have the coefficients of various 's will not interfere with each other, so it remains to choose the coefficients . Now choose with size sufficiently rapidly decreasing so the converges on compacts (and for the rational case, it is easy to make it transcendence --- indeed in this case this is guaranteed by the arbitrarily long sting of zeros).

There is nothing special about or in this problem, except they are countable subrings with an element inside the punctured unit disc, so we can replace them with, e.g. or the algebraic closure of .

Ah I see, I was trying to use the interpolation to produce a sequence, but I couldn't see the bit with the for convergence. Also, I like that bump construction. Nice problem!

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