Timeline for Behavior of $n^\alpha \sin^{\circ\, n}(n^{-\alpha}x)$
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Oct 4, 2021 at 10:17 | history | edited | YCor | CC BY-SA 4.0 |
added reference info
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| Oct 4, 2021 at 8:50 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
corrected the derivation of the asymptote for \alpha<1/2, which now agrees well with the numerics.
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| Oct 4, 2021 at 2:23 | comment | added | Gerry Myerson | @Yaakov, you were not the only one with misgivings. See meta.mathoverflow.net/questions/5150/… where the question | |
| Oct 3, 2021 at 18:24 | comment | added | username | $$x-\frac{x^3}{6}<\sin(x)<x$$ for $x>0$, so $$\sin^{ck}(x)-\frac16 (\sin^{ck}(x))^3 < \sin^{c(k+1)}(x) < \sin^{ck}(x),$$ which in turn yields $$ \sin^{ck}(x)-\frac{x^3}{6}<\sin^{c(k+)1}(x)<x, \textrm{ and } x-n\frac{x^3}{6}<\sin^{cn}(x)<x $$ So writing $v_n= n^\alpha \sin^{cn}\left(n^{-\alpha}x\right)$, we have $$ x- n^{1-2\alpha} \frac{x^3}{6} < v_n(x) < x $$ So when $\alpha > \frac12$, $v_n\to x$. | |
| Oct 3, 2021 at 14:24 | history | wiki removed | Stefan Kohl♦ | ||
| Oct 3, 2021 at 14:24 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Oct 3, 2021 at 12:48 | comment | added | Yaakov Baruch | At first I had some misgivings about one changing a question and then answering it, but looking up the paper I can see that in this case it is both justified and interesting. | |
| Oct 3, 2021 at 12:19 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
asymptotics for alpha < 1/2
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| Oct 3, 2021 at 10:17 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
one more plot added
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| Oct 3, 2021 at 9:46 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
added 9 characters in body
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| Oct 3, 2021 at 7:16 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
added 236 characters in body
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| Oct 3, 2021 at 6:47 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
added graph, removed the trivial case, which is well answered in the comments
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| Oct 2, 2021 at 19:54 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
deleted 3 characters in body
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| Oct 2, 2021 at 19:52 | history | edited | LSpice | CC BY-SA 4.0 |
Name of arXiv reference
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| Oct 2, 2021 at 19:49 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
added 90 characters in body
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| Oct 2, 2021 at 19:44 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 2, 2021 at 19:34 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |