Timeline for Approximating smooth functions with polynomials subject to constraints.
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 12, 2011 at 14:19 | comment | added | David Carchedi | Thank you to everyone who has helped me think about this. It appears that I need something much stronger than what I have said for my purposes, and you have made me realize this. | |
| Dec 9, 2011 at 19:15 | answer | added | Vadim Alekseev | timeline score: 2 | |
| Dec 9, 2011 at 6:34 | answer | added | Robert Israel | timeline score: 1 | |
| Dec 8, 2011 at 21:23 | comment | added | Will Sawin | For "close" you need a topology. The only topology that makes sense here is the $x$-adic topology, I think. (So $x^n$ divides $h\circ F-h\circ G$ for some large $n$) Do you agree? Also, do you want to approximate near a point? | |
| Dec 8, 2011 at 20:13 | comment | added | David Carchedi | @Robert: Good point. So I suppose it is unreasonable to demand that these polynomials literally satisfy the equation. Instead (as I remarked in parentheses) one can ask can we approximate $h$ by polynomials such that $h \circ F$ and $h \circ G$ are "close". | |
| Dec 8, 2011 at 20:04 | comment | added | Robert Israel | The condition $h \circ F = h \circ G$ is a functional equation, which may have no nonconstant polynomial solutions. For example, consider $n=m=1$, $F(x) = x$, $G(x) = x+1$, where the functional equation says $h$ is periodic with period 1. | |
| Dec 8, 2011 at 19:30 | comment | added | David Carchedi | @Yemon: For me though, it's ok not to even have a norm. I just need a reasonably functorial topology on the space of smooth functions for which I can find a convergent sequence of polynomials satisfying the properties I listed. | |
| Dec 8, 2011 at 19:25 | comment | added | David Carchedi | @Yemon: Sadly, I have to let $h$ be an arbitrary smooth function, so I cannot assume it's bounded. | |
| Dec 8, 2011 at 19:19 | comment | added | Yemon Choi | @David: since your domain is non-compact, is $h$ supposed to be bounded? If not, then you will need to have a weight in the norm, which makes life tricky. For instance, what are you hoping for in the case $m=n=1$? | |
| Dec 8, 2011 at 18:56 | comment | added | David Carchedi | @Yemon: I want to leave that vague for now. If this can be done in some norm I am interested. | |
| Dec 8, 2011 at 18:52 | comment | added | Yemon Choi | In what norm are you approximating? | |
| Dec 8, 2011 at 18:43 | history | asked | David Carchedi | CC BY-SA 3.0 |