Timeline for Positive definite function on the upper half line and its integral expression
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 12, 2023 at 8:36 | review | Suggested edits | |||
| Aug 12, 2023 at 15:04 | |||||
| Jul 25, 2023 at 0:51 | comment | added | Thomas Kojar | To get the form you particularly mentioned, it would seem that you also need radial, which in 1d you get for free. So a suggestion is to try to mimic the proofs that positive-def+radial = completely monotone for your particular class of functions with x+y instead. | |
| Jul 25, 2023 at 0:48 | comment | added | Thomas Kojar | This nonnegative is crucial to Hermitian inner product. | |
| Jul 25, 2023 at 0:46 | comment | added | Thomas Kojar | So I am taking a look whether you can at least get Bochner back. I am looking at the Folland proof typed here too online individual.utoronto.ca/jordanbell/notes/bochnertheorem.pdf. It seems to me that you still get Theorem 3 because I think you can just skip the very last inequality and you get nonnegative where he has $\phi(x+y)$. | |
| Jul 25, 2023 at 0:34 | comment | added | user509119 | It is not $f(x_k-x_l)$ but $f(x_k+x_l)$. | |
| Jul 25, 2023 at 0:25 | comment | added | user509119 | Thank you for your referece, but the definition of positive definiteness I adopt differs from the one. | |
| Jul 25, 2023 at 0:16 | comment | added | Thomas Kojar | you have Bochner's theorem at least math.iit.edu/~fass/603_ch2.pdf. Also check out Theorem 2.4.2 and Theorem 2.5.3 if you also have radial. | |
| Jul 25, 2023 at 0:09 | history | asked | user509119 | CC BY-SA 4.0 |