Timeline for Non-analytic function with convergent Taylor series everywhere
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 10, 2020 at 13:12 | comment | added | Dave L Renfro | The Math Forum URLs I posted in my 22 November 2011 comment no longer work, but these from the internet archive still work: Part 1 and Part 2 | |
| Mar 9, 2012 at 22:30 | answer | added | Bazin | timeline score: 3 | |
| Nov 23, 2011 at 15:32 | comment | added | Bruce Blackadar | After I finally read some definitive history on this subject such as Dave's survey, what strikes me is how consistently many mathematicians (now myself included) have been ignorant of previous work in this area over a rather long time. I don't offhand recall any topic, at least in analysis, where basic examples and results have been rediscovered, reproved, and republished so often, and I would guess that even today many mathematicians would not know about this work. | |
| Nov 22, 2011 at 21:22 | comment | added | Bruce Blackadar | OK, thanks, I guess I didn't do my homework on this one. | |
| Nov 22, 2011 at 20:58 | comment | added | Charlie Frohman | @fedja It's the kind of question I would ask. | |
| Nov 22, 2011 at 19:39 | comment | added | Dave L Renfro | I posted a fairly extensive survey on this topic in May 2002 at mathforum.org/kb/message.jspa?messageID=387148 and mathforum.org/kb/message.jspa?messageID=387149 | |
| Nov 22, 2011 at 18:53 | comment | added | Pietro Majer | wow ! | |
| Nov 22, 2011 at 18:03 | comment | added | fedja | @Charlie: OK, I'll be giving a graduate course in complex analysis next semester. I'll assign it as homework and let you know the result :). Remind me if I forget. @Bruce: The set $A_{m,n}$ of points $x$ for which $|f^{(k)}(x)|\le mk!n^k$ for all $k$ is closed and contains no interval. Ergo... | |
| Nov 22, 2011 at 16:29 | comment | added | Charlie Frohman | This question would be a reasonable homework question in a course where you were broaching the concept of analyticity. | |
| Nov 22, 2011 at 16:22 | comment | added | Charlie Frohman | @Ali if the Taylor series doesn't converge to the function. In Gerald Edgar's example he didn't define his function at $x=0$ to be $0$. The Taylor series at $0$ is then the $0$ series which is not equal to the function he gave. | |
| Nov 22, 2011 at 16:03 | comment | added | user16974 | I don't understand the whole situation: how can a function with a convergent Taylor series expansion at every point and with positive radius of convergence be non analytic? | |
| Nov 22, 2011 at 15:51 | comment | added | Gerald Edgar | How about taking an example (like $\exp(-1/x^2)$) with this property at one point only, then doing a series with translates by the rationals and coefficients going to zero fast enough? | |
| Nov 22, 2011 at 14:49 | history | asked | Bruce Blackadar | CC BY-SA 3.0 |