Timeline for Approximate simple function $f$ by a sequence of continuous functions on $\mathbb{R}^d$ such that $\|f_n\|_\infty\leq \|f\|_\infty$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 26 at 14:24 | answer | added | Pietro Majer | timeline score: 6 | |
| Jul 26 at 14:24 | comment | added | Pietro Majer | OK, thank you, I wasn't sure I had understood the question. | |
| Jul 26 at 14:17 | comment | added | Aleksei Kulikov | @PietroMajer huh, this is way simpler and much more elegant than the monstrosity that I have created. Why don't you post this as an answer? | |
| Jul 26 at 9:09 | comment | added | Pietro Majer | Yes, I asked just for being sure the final goal of the OP was not just approximating by $f_n$ below $f.$ Another possibility is an approximation of the identity via convolution; by linearity it smoothes the $1_{\Delta_i}$ as well, and of course $\|\phi_\epsilon*f\|_\infty \le \|f\|_\infty$ | |
| Jul 26 at 6:59 | comment | added | Aleksei Kulikov | @PietroMajer because this will do some unspeakable horrors to the functions $f_k^m$. I interpreted the goal (and the difficulty) of the question being that we want to preserve the existence of the functions $f_k^m$ while adding the extra norm condition. | |
| Jul 26 at 6:51 | comment | added | Pietro Majer | Why don't you truncate $f_n$ at $M:=\|f\|_\infty$, and consider $(f_n\wedge M)\vee(-M)$ that has the same limit $f$ of $f_n$? | |
| Jul 26 at 6:50 | answer | added | Aleksei Kulikov | timeline score: 3 | |
| Jul 26 at 6:32 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing
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| Jul 26 at 6:08 | comment | added | mathlover | yes, want almost everywhere | |
| Jul 26 at 6:05 | comment | added | Aleksei Kulikov | I would also assume that you want convergence almost everywhere, not everywhere, right? | |
| Jul 26 at 6:03 | comment | added | mathlover | $\Delta_i$ are measurable sets with $|\Delta_i|<\infty$ | |
| Jul 26 at 6:00 | comment | added | Aleksei Kulikov | Is $\Delta_i$ an open set, measurable set or something else? | |
| Jul 26 at 5:49 | history | asked | mathlover | CC BY-SA 4.0 |