Timeline for $L^p$ norm means
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 8, 2023 at 11:39 | answer | added | Venkata Karthik Bandaru | timeline score: 0 | |
| Sep 9, 2014 at 12:49 | comment | added | Liviu Nicolaescu | Section 1.8 from the book Special functions by Andrews, Askey and Roy might help. It describes Dirichlet's methods mentioned by WillJagy and gives several useful consequences. | |
| Sep 9, 2014 at 9:21 | comment | added | Bill Johnson | Two papers that might be relevant are: G. Schechtman and J. Zinn, On the volume of the intersection of two $L^n_p$ balls, Proc. A.M.S. 110 (1990), 217–224. G. Schechtman and M. Schmuckenschlager, Another remark on the volume of the intersection of two $L^n_p$ balls, GAFA Seminar 89/90, Lecture Notes in Math., Vol 1469, 174–178, Springer (1991). | |
| Sep 9, 2014 at 9:10 | answer | added | Guillaume Aubrun | timeline score: 8 | |
| Sep 9, 2014 at 1:55 | answer | added | Christian Remling | timeline score: 2 | |
| Sep 9, 2014 at 1:34 | answer | added | Will Jagy | timeline score: 3 | |
| Sep 9, 2014 at 1:26 | comment | added | Will Jagy | It's in his collected works, books.google.com/… | |
| Sep 9, 2014 at 1:17 | comment | added | Anthony Quas | I think @YemonChoi is exactly right here (for the asymptotics of large $n$). For the large $q$ asymptotics, I think the normals are useful again: probably it is something like $\sqrt{2\log n}$ as $q\to\infty$. | |
| Sep 9, 2014 at 1:11 | comment | added | Igor Rivin | @WillJagy Erm, where do I find the original? For that matter, where is it in W&W? With bated breath... | |
| Sep 9, 2014 at 1:02 | comment | added | Will Jagy | Might help, Dirichlet invented a way to integrate any polynomial on the body $$ x_1^{a_1} + \cdots x_n^{a_n} \leq 1, $$ it is in Whittaker and Watson but more clear in the original. | |
| Sep 9, 2014 at 0:34 | comment | added | Yemon Choi | Also, for the $p=2$ case I am tempted to make a first guess (not a proper derivation of the correct asymptotic) by looking at a Gaussian vector with i.i.d. entries that are $N(0,n^{-1/2})$, this is "mostly" concentrated on the unit sphere and the expected $L^q$-norm would seem to have some closed form that allows for decent estimates asymptotic in $n$. But as I said this may be a case of working out the asymptotics for something different from what was intended. | |
| Sep 9, 2014 at 0:27 | comment | added | Yemon Choi | In that case, doesn't it reduce to the $p=2$ case by scaling? (It is 1am here so forgive me if I have missed something obvious) | |
| Sep 9, 2014 at 0:26 | comment | added | Igor Rivin | @YemonChoi Yes, it is the induced measure, though I have to admit that I was secretly thinking $p=2$ - that case might well be much easier. | |
| Sep 9, 2014 at 0:15 | comment | added | Yemon Choi | A naive question: what's your choice of measure on $S_p^{n-1}$ when $p\neq 2$? Is it just the image of the uniform measure on the Euclidean sphere under the natural homeo from that sphere to the $L^p$-version? | |
| Sep 9, 2014 at 0:03 | history | asked | Igor Rivin | CC BY-SA 3.0 |