Timeline for Functions with derivatives growing at rate $r>0$
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| May 18 at 14:28 | vote | accept | Math_Newbie | ||
| May 16 at 9:21 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| May 16 at 5:04 | vote | accept | Math_Newbie | ||
| May 16 at 3:28 | answer | added | Willie Wong | timeline score: 3 | |
| May 16 at 2:45 | comment | added | Willie Wong | Eh, do you really want $k\in \mathbb{N}$ including $0$? Because this would require $|f(x)| \lesssim 0$ which means $f \equiv 0$. Presumably you want $0$ to be excluded. | |
| May 15 at 21:46 | comment | added | Math_Newbie | @ChristianRemling That's a fair point. I also edited the post to ask about the case where we only max over a compact domain e.g. the interval. Then polynomials are clearly included... | |
| May 15 at 21:45 | history | edited | Math_Newbie | CC BY-SA 4.0 |
added 169 characters in body
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| May 15 at 21:33 | comment | added | Christian Remling | I think the argument from the linked MSE post still works (without having checked all the fine details): if $\|f^{(k)}\|_{\infty}\le k^N$, then $f$ is entire of exponential type $\le 1$. By Paley-Wiener, $\widehat{f}$ is supported by $[-1,1]$, and this should give $\|f^{(k)}\|_{\infty}\le \|f\|_{\infty}$, so that there are no additional functions. | |
| May 15 at 20:21 | comment | added | Christian Remling | Of course, we can also write $\le k^r$ instead of $\lesssim k^r$ and it's still the same question (since we can replace $f$ by $cf$). | |
| May 15 at 19:57 | comment | added | Math_Newbie | Exactly what I meant :) | |
| May 15 at 19:57 | comment | added | Abdelmalek Abdesselam | most likely $\exists C>0$, $\forall k\ge 0$, the sup is $\le Ck^r$. | |
| May 15 at 19:40 | comment | added | Iosif Pinelis | What do you mean by $\lesssim k^r$? | |
| May 15 at 19:07 | comment | added | Math_Newbie | sorry I mean $s->k$ (I sometimes use "s" for "smoothness"). So I'm wondering what classes of functions are included when we fix a "polynomial-esque" growth rate for the maximal $k^{th}$ order of its derivatives. | |
| May 15 at 19:05 | history | edited | Math_Newbie | CC BY-SA 4.0 |
edited body
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| May 15 at 18:59 | history | edited | LSpice | CC BY-SA 4.0 |
Name of nice post
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| May 15 at 18:12 | history | asked | Math_Newbie | CC BY-SA 4.0 |