Timeline for Determining polynomial approximations of piecewise constant functions
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 18, 2021 at 15:16 | answer | added | Joseph Van Name | timeline score: 1 | |
| Sep 18, 2021 at 13:40 | comment | added | Joseph Van Name | If there any reason you want pointwise convergence instead of uniform convergence on compact sets? | |
| Sep 18, 2021 at 12:31 | answer | added | Alexandre Eremenko | timeline score: 3 | |
| Sep 17, 2021 at 19:10 | vote | accept | user918212 | ||
| Sep 17, 2021 at 19:05 | answer | added | Pietro Majer | timeline score: 3 | |
| Sep 17, 2021 at 17:53 | comment | added | user44143 | Since you have Heaviside approximations already, let $H_\epsilon(x)$ be a polynomial which is between $0$ and $\epsilon$ when $ -1<x<-\epsilon$, between $0$ and $1$ when $-1<x<1$, and between $1-\epsilon$ and $1$ when $\epsilon<x<1$. Now assume WLOG that $t_1=0$, $t_m=1$. Then let $f_1(x)=a_1$ $$f_{i+1}(x)=(1+\frac{a_{i+1}}{a_i}H_{\epsilon/m}(x-t_i))f_i(x).$$ I claim that $f_m(x)$ is now within $\epsilon$ on $f(x)$ except on a set of measure $2m \epsilon$, so taking the $f_m$ as $\epsilon\to 0$ gives the desired sequence of approximations. | |
| Sep 17, 2021 at 17:07 | history | asked | user918212 | CC BY-SA 4.0 |