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 September 11th, 2016, 05:43 PM #1 Senior Member   Joined: Sep 2016 From: USA Posts: 383 Thanks: 207 Math Focus: Dynamical systems, analytic function theory, numerics 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? Thanks from raul21
 September 12th, 2016, 09:37 AM #2 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,364 Thanks: 100 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
 September 12th, 2016, 10:53 PM #3 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,364 Thanks: 100 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.
 September 13th, 2016, 08:15 PM #4 Senior Member   Joined: Sep 2016 From: USA Posts: 383 Thanks: 207 Math Focus: Dynamical systems, analytic function theory, numerics 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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