Timeline for Law of a step function and its generalization to two dimensions on an appropriate spaces
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2020 at 8:14 | vote | accept | Mark | ||
| Mar 10, 2020 at 10:45 | comment | added | Mark | @IosifPinelis, thank you for your detailed answer. I don't know if I will end up using your space $\tilde B_2$. But that is great example. It was recommended me to use space such as $\tilde B_2:=C([0,T];G)$ with appropriate $G$ and with appropriate convergence. On bounded domains $L^{\infty} \subset L^1 \subset \mathcal{M} \subset \mathcal D '$. So I thinkI will use space $\tilde{B}_2:=C([0,T];\mathcal{M})$ i.e. $G=\mathcal{M}$ and of course with weak convergence of measures (i.e. $C_{w}([0,T];\mathcal{M})$ on that big list in my MSE question). Hope that I am not mistaken. Thanks again. | |
| Mar 10, 2020 at 9:50 | comment | added | Mark | @LSpice, thanks for the correction. It is one of those things I didn't learn correctly from the start, and now I make that mistake from time to time. | |
| Mar 9, 2020 at 22:07 | comment | added | LSpice |
@Mark, $\mathcal D^{'}$ $\mathcal D^{'}$ (as in 1 2) sets the prime too high. You want $\mathcal D'$ $\mathcal D'$ (as in 3).
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| Mar 9, 2020 at 22:05 | comment | added | Iosif Pinelis | @Mark : I have added a response to your latest comment. | |
| Mar 9, 2020 at 22:04 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 3 characters in body
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| Mar 9, 2020 at 21:41 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 3 characters in body
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| Mar 9, 2020 at 18:33 | comment | added | Mark | Thanks for the correction on $BV$ and $\mathcal{D}^{'}$. For $u_1$ spaces such as $\mathcal{M}$ and $L^p, p\in[0,\infty]$ sounds good for application. However I am not sure for the $u_2$. For it I need some Banach space-valued spaces. Last year I asked question on MSE for the function $u_2$ (edited today) (math.stackexchange.com/questions/3254479/…). There is a big list there at the end of it. I think that at least some of those Banach space-valued spaces could work. Am I wrong? And thanks again. | |
| Mar 9, 2020 at 18:18 | comment | added | Iosif Pinelis | @Mark : Yes, you can use many other spaces instead of $L^\infty$. However, for $BV$ and $\mathcal D'$, you need to assume that $D$ is open and also replace $[0,T]$ by $(0,T)$. | |
| Mar 9, 2020 at 18:10 | comment | added | Mark | Thank you for the answer. For my problem it doesn't important what $u_1(c_1)$, $u_2(c_2 \cdot t, t)$ is. You could take $u_R$ for both of them. My guesses for $u_1$ would be $L^{\infty}(D)$ too, but also $BV(D)$ - space of functions with bounded variation, $\mathcal{M}(D)$ - space of Radon measures and $\mathcal{D}^{'}(D)$ - space of distributions. I am not sure if I am right. For the function $u_2$ the space $L^{\infty}(D \times [0,T])$ sounds good. I guess also that the spaces $BV(D \times[0,T])$, $\mathcal{M}(D \times [0,T])$, $\mathcal{D}^{'}(D \times [0,T])$ could work too? | |
| Mar 9, 2020 at 15:37 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |