Timeline for Bounds on polynomial values
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 10, 2017 at 14:49 | answer | added | user111 | timeline score: 3 | |
| Sep 8, 2017 at 19:12 | vote | accept | T. Amdeberhan | ||
| Sep 8, 2017 at 7:18 | comment | added | Fedor Petrov | @NateEldredge we may use instead $|\sum a_kf_k(x)|\leqslant \sqrt{\sum f_k^2(x)}$, which is sharp for appropriate $a_k$'s. So, the problem is to estimate the sum of squares of Legendre polynomials. | |
| Sep 8, 2017 at 4:10 | comment | added | Nate Eldredge | ... and in light of Igor Rivin's answer, the latter of these is exactly what Polya and Szego's solution does. Problem 93 gives $\|f_k\|_\infty^2 = {(2k+1)/2}$. (Here $\|\cdot\|_\infty$ is of course the sup over $[-1,1]$.) | |
| Sep 7, 2017 at 22:15 | answer | added | Igor Rivin | timeline score: 10 | |
| Sep 7, 2017 at 22:07 | comment | added | Nate Eldredge | @IgorRivin: All I mean is that since they are orthonormal, we can write $f = \sum_{k=0}^n a_k f_k$ where $\sum_{k=0}^n a_k^2 = 1$, so for instance we get trivial bounds like $\|f\|_{\infty} \le \sum_{k=0}^n \|f_k\|_{\infty}$ or $\|f\|_\infty^2 \le \sum_{k=0}^n \|f_k\|_\infty^2$. | |
| Sep 7, 2017 at 22:05 | comment | added | Igor Rivin | @NateEldredge Why are legendre polynomials extremal? | |
| Sep 7, 2017 at 19:49 | comment | added | Fedor Petrov | Alternative reformulation: $\sum_{k=0}^n P_k^2\leqslant \frac{n+1}2$ on $[-1,1]$, where $P_i$ are orthonormal Legendre polynomials. | |
| Sep 7, 2017 at 19:01 | answer | added | Christian Remling | timeline score: 2 | |
| Sep 7, 2017 at 18:21 | comment | added | Steve Huntsman | @NateEldredge math.stackexchange.com/questions/417999 | |
| Sep 7, 2017 at 16:33 | comment | added | Nate Eldredge | You would get some sort of trivial bound if you knew the sup norm of the Legendre polynomials, which I don't know but I assume must be known. I wonder if it recovers your bound? | |
| Sep 7, 2017 at 16:15 | history | asked | T. Amdeberhan | CC BY-SA 3.0 |