Timeline for When can a family of polynomials get a weight function to be made orthogonal?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 28, 2012 at 10:11 | vote | accept | Wolfgang | ||
| Jan 27, 2012 at 23:56 | comment | added | Noam D. Elkies | Henry is right, and I apologize for the typo. | |
| Jan 27, 2012 at 23:21 | comment | added | Henry Cohn | (But you're right that this isn't a particularly useful condition for proving that polynomials are orthogonal. I view Favard's theorem as having mainly psychological value: if you observe a suitable recurrence experimentally, then you really ought to look for an orthogonality proof to explain it. It's sometimes possible to prove the recurrence directly and work from there, but this is generally not the most flexible or illuminating approach.) | |
| Jan 27, 2012 at 23:18 | comment | added | Henry Cohn | Oops, I started writing my answer before this comment appeared. The three-term recurrence should be $xP_{n}(x) = P_{n+1}(x) + A_n P_n(x) - B_n P_{n-1}(x)$ with $B_n>0$, and then it actually implies that the roots are interlaced. | |
| Jan 27, 2012 at 23:13 | answer | added | Henry Cohn | timeline score: 10 | |
| Jan 27, 2012 at 23:06 | comment | added | Noam D. Elkies | Orthogonal polynomials must satisfy a three-term recursion $x P_{n+1}(x) = A_n P_n(x) - B_n P_{n-1}(x)$ and have real interlaced roots. Not clear how to test these necessary conditions from a generating function or contour integral, let alone give sufficient conditions. | |
| Jan 27, 2012 at 22:35 | history | asked | Wolfgang | CC BY-SA 3.0 |