Timeline for Positive eigenfunctions of the discrete Fourier transform
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 15 at 18:17 | comment | added | Itay | @TerryTao Thank you! | |
| Aug 15 at 17:15 | comment | added | Terry Tao | Getting the result to be non-negative and maximized at the origin may require some further tricks and normalizations, e.g., restricting $f$ to first be non-negative and positive definite. | |
| Aug 15 at 17:13 | comment | added | Terry Tao | Since ${\mathcal F}^4 = I$, you can average any function $f$ with its first three Fourier transforms ${\mathcal F}(f)$, ${\mathcal F}^2(f)$, ${\mathcal F}^3(f)$ and normalize, and this is a complete description of the eigenfunctions of eigenvalue 1 (the eigenfunctions for the other three eigenvalues $i, -1, -i$ are produced similarly). If $f$ is even then ${\mathcal F}^2(f)=f$ and one just needs to average $f$ and ${\mathcal F}(f)$ as before. | |
| Aug 15 at 15:35 | comment | added | Itay | @TerryTao Yes, I should have written that as a trivial example. You can actually take the average of every subgroup and its dual and get the same. Are there any other known constructions beyond convex combinations of such? | |
| Aug 15 at 14:32 | comment | added | Terry Tao | The average of the (suitably normalized) delta function and constant function. | |
| Aug 15 at 9:47 | history | asked | Itay | CC BY-SA 4.0 |