Timeline for Other expansion for positive Taylor expansion
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 27, 2023 at 13:30 | comment | added | Alexandre Eremenko | The signs of $a_k$ do not matter, what does matter in my remark is $b_k>0$. | |
| Aug 26, 2023 at 21:10 | vote | accept | NancyBoy | ||
| Aug 26, 2023 at 18:38 | history | became hot network question | |||
| Aug 26, 2023 at 13:03 | vote | accept | NancyBoy | ||
| Aug 26, 2023 at 21:10 | |||||
| Aug 26, 2023 at 12:00 | vote | accept | NancyBoy | ||
| Aug 26, 2023 at 13:02 | |||||
| Aug 26, 2023 at 11:59 | comment | added | NancyBoy | Thabk you Alexander but here there is only $f>0$, we don't know the sign of the $a_k$. | |
| Aug 26, 2023 at 11:53 | comment | added | Alexandre Eremenko | The answer is trivial: take $b_0=1, b_k=0$ for $k\geq 2$ and $\beta_0=f$. You have to specify what functions $\beta_k$ are allowed. With exponential functions this is in general not possible, since in this case we will have $f^{(n)}>0$ for all $n$, so not all positive functions can be expanded into a series of positive exponentials. | |
| Aug 26, 2023 at 11:20 | answer | added | Michał Jan | timeline score: 5 | |
| Aug 26, 2023 at 10:23 | comment | added | NancyBoy | Arh, thank you for these information ! | |
| Aug 26, 2023 at 10:21 | comment | added | Michał Jan | Ah, sadly it won't. Inverse Laplace transform does not preserve positivity mathoverflow.net/questions/383996/… | |
| Aug 26, 2023 at 10:17 | comment | added | NancyBoy | Thank you for your answer ! My problem is that the $a_k$ are not necessary positive but $f$ is positive. I want to have another expression with only positive terms. Do you think that Laplace transform will work ? | |
| Aug 26, 2023 at 10:14 | comment | added | Michał Jan | If it's enough for this to work for $x>0$, and you change the sum for an integral, then this becomes Laplace transform $f(x)=\int_0^\infty dk b(k) \exp(-kx)$. For positive $b(k)$, $f(x)$ is of course positive, but I don't know if converse is true. | |
| Aug 26, 2023 at 9:03 | history | asked | NancyBoy | CC BY-SA 4.0 |