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September 11th, 2016, 06:43 PM  #1 
Senior Member Joined: Sep 2016 From: USA Posts: 241 Thanks: 126 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 \[ f(x) = \sum_{n\geq 0} a_n (xc)^n \] 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? 
September 12th, 2016, 10:37 AM  #2 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,214 Thanks: 91 
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 11:17 AM. 
September 12th, 2016, 11:53 PM  #3 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,214 Thanks: 91 
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 lookup 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 abstractspeak I suspect would be of very limited interest. 
September 13th, 2016, 09:15 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 241 Thanks: 126 Math Focus: Dynamical systems, analytic function theory, numerics 
The norm is the weighted $\ell^1$ norm defined by \[ u = \sum_{n \geq 0} u_nr^n. \] 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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