Timeline for Coarea-like formula for BV functions (not their derivative)
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Apr 20, 2019 at 7:59 | comment | added | Skeeve | @Riku One could as an analogous question for complex valued functions (or for vector valued ones). So maybe you are looking for something like $f = \int_0^\infty \frac{f}{|f|} \chi_{\{|f|>t\}} \, dt$? (Of course we should define $\frac{f}{|f|}=0$ whenever $|f|=0$.) | |
| Apr 19, 2019 at 18:34 | comment | added | Riku | @WillieWong In that case get $f(x)=\int _{0}^{\infty }1_{L(f,t)}(x)\,dt + \int _{\infty}^{0}1_{E(f,t)}(x)\,dt,$ where $L(f,t)=\{y\in {\mathbb {R}}^{n}|f_+(y)\geq t\}$ and $E(f,t)=\{y\in {\mathbb {R}}^{n}|f_-(y)\geq t\}$. Can it be written in a better way as an integral over $\mathbb R$? | |
| Apr 19, 2019 at 18:22 | comment | added | Willie Wong | Write $f = f_+ - f_-$? | |
| Apr 19, 2019 at 18:21 | comment | added | Riku | @WillieWong That sounds great. How can it be generalized if $f$ is not necessarily non-negative? | |
| Apr 19, 2019 at 18:18 | comment | added | Willie Wong | What's wrong with en.wikipedia.org/wiki/Layer_cake_representation ? | |
| Apr 19, 2019 at 18:10 | history | asked | Riku | CC BY-SA 4.0 |