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September 11th, 2016, 05:43 PM   #1
SDK
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Is this a principal ideal domain?

Every univariate real analytic function has an expansion of the form
\[
f(x) = \sum_{n\geq 0} a_n (x-c)^n
\]
at each $c$ in its domain of analyticity.

Each of these series converges uniformly and absolutely on some interval, commonly called the radius of convergence. It is trivial to show that if we fix a radius of convergence, $\tau > 0$, then the converse statement holds. Namely, if $a \in X$ where
\[
X = \{ \{a_n\}_n \subset \mathbb{R} : \sum_{n \geq 0} |a_n| \tau^n < \infty \},
\]
then $a$ defines an analytic function on $(-\tau, \tau)$. Moreover, defining the norm on $X$ in the obvious way and noting that it is closed under multiplication (multiplication given by Cauchy products), this space is a Banach algebra.

Question:
Is this Banach algebra a principal ideal domain?
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September 12th, 2016, 09:37 AM   #2
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What is {$\displaystyle a_{n}$}$\displaystyle _{n}$?

Edit:
and what is the norm of all functions f(x) above with the same radius of convergence, given x<r, ie, can you define the norm of a function defined on an open interval?

Last edited by zylo; September 12th, 2016 at 10:17 AM.
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September 12th, 2016, 10:53 PM   #3
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OK
a={a1,a2,..........,a\infty} with domain specified by f(x) and r fixed.

The norm is usual vector norm.

How does a define an analytic function on (-r,r)?

Beyond that I thought it might be fun to drill through the definitions and learn something with a nice discussion. Constructing a logical structure of meaningful definitions by look-up turned out to be difficult and not worth time and effort, for me.

If OP wanted to do it, that would probably be of general interest. A solution in abstract-speak I suspect would be of very limited interest.
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September 13th, 2016, 08:15 PM   #4
SDK
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The norm is the weighted $\ell^1$ norm defined by
\[
||u|| = \sum_{n \geq 0} |u_n|r^n.
\]
This defines a norm whenever $x$ defines an analytic function on $(-r,r)$. The meaning of this is that the series converges uniformly for any $x \in (-r,r)$. Clearly any sequence in this space also defines an analytic function.

In fact this is a Banach algebra which prompts the question. I'm not an algebraist and this question was recently asked about whether this algebra is principally generated. I'm 99% sure the answer is no but I can't come up with a proof.
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