Timeline for Bounds on Legendre polynomials on the complex plane
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 9, 2017 at 21:45 | comment | added | user111 | The growth of $\kappa_n$ is exponential outside $[-1,1]$ while it is only polynomial on $[-1,1]$ which is consistent with the fact that all the zeros of the $P_n$'s lie on that interval. So, for your inequality, this is the "critical" set. Hope that helps. | |
| Sep 9, 2017 at 19:26 | vote | accept | Doanh Doanh | ||
| Sep 9, 2017 at 19:26 | comment | added | Doanh Doanh | Thanks for the replies. I get the part about $\kappa$. My question was that the Bernstein inequality in mathoverflow.net/questions/151978/… only seems to work for real x. | |
| Sep 9, 2017 at 14:07 | comment | added | user111 | Does $z+\sqrt{z^2-1}$ vanish outside of $[-1,1]$ ? | |
| Sep 9, 2017 at 12:15 | comment | added | Doanh Doanh | But it doesn't work for x not real, does it? | |
| Sep 8, 2017 at 5:23 | comment | added | user111 | yes, the constants $C_x$, $x\in(-1,1)$, are bounded below by some positive constant $C$ independent of $x$. For a bound on the derivative, see e.g. mathoverflow.net/questions/151978/…. | |
| Sep 7, 2017 at 20:51 | comment | added | Doanh Doanh | For a fixed point x, then using the limits that you mentioned, we get the bound $\kappa_n(x) \gg n^a$. I need a bound that holds for every x, i.e. the implied constant does not depend on x. Other than the limits, do you know any more precise estimates? Btw, do you know any bound like $|Q'(z)| << n$ for $z$ in the question? Thanks a bunch! | |
| Sep 6, 2017 at 20:52 | history | answered | user111 | CC BY-SA 3.0 |