Timeline for approximation of rational functions
Current License: CC BY-SA 3.0
7 events
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| Mar 22, 2017 at 22:04 | comment | added | Trevor J Richards | Though now I see that you want some control on the speed of convergence, forcing $\epsilon'$ to have a special form. This is a more complicated question than I realized. By the way, I do not see how the condition $Re(q/\widehat{q})>0$ is related to $|q-\widehat{q}|<\epsilon'$. Could you explain that? | |
| Mar 22, 2017 at 22:00 | comment | added | Trevor J Richards | Hmmm, this sort of has the flavor of a converse to Runge's theorem: If a sequence of rational functions $\widehat{r_i}$ (in your example these are choices of $\widehat{p}/\widehat{q}$) converges to a meromrophic function $r$ (in your example $p/q$) on a compact set $K$ (in your example the unit circle), and the degree of all of the rational functions $r_i$ are bounded by some number $n$, then the zeros of $r_i$ must converge and the poles of $r_i$ must converge. I will think about it. Have you gotten a solution to the problem you posted yourself yet? | |
| Feb 21, 2016 at 1:36 | comment | added | Alex Wenxin Xu | Ah.. yeah, just assume that $p$ and $\hat{p}$ has leading coefficient 1. This should be rather a minor thing though. | |
| Feb 19, 2016 at 21:46 | comment | added | Robert Israel | How do these conditions rule out, e.g., the possibility that $\hat{p} = 2 p$ and $\hat{q} = 2 q$? | |
| Feb 19, 2016 at 18:39 | history | edited | Alex Wenxin Xu |
edited tags
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| Feb 19, 2016 at 2:59 | history | edited | Alex Wenxin Xu | CC BY-SA 3.0 |
added 152 characters in body
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| Feb 19, 2016 at 2:29 | history | asked | Alex Wenxin Xu | CC BY-SA 3.0 |