Timeline for Max of Fourier transform?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| May 29, 2012 at 21:38 | answer | added | Bazin | timeline score: 2 | |
| May 29, 2012 at 13:34 | comment | added | H A Helfgott | Obviously I am talking about a function that is not positive everywhere. Say somebody gives you a function that is piecewise defined as segments of the form $x\mapsto C/x$ (say) and then you get a distribution f by summing that function to three point masses, one or two of them negative. How do you go about finding $\max_{\alpha\in \mathbb{R}} |\widehat{f}(\alpha)|$ (rigorously and to a certain accuracy)? | |
| May 29, 2012 at 12:01 | comment | added | Willie Wong | What do you mean by something "better"? For positive real valued function $f$, you necessarily have $\hat{f}(0) = |f|_1$, so the inequality is sharp. (The same argument also shows that in the class of distributions which can be decomposed into a sum of a smooth function and three point masses, the inequality is sharp. So you probably need to say more.) Do you mean you want a bunch of statements of the form "If $f$ satisfies conditions blah and blah then we can improve the inequality to (some inequality)"? | |
| May 29, 2012 at 11:22 | comment | added | H A Helfgott | Just to give an idea - the f I am working with is a nice smooth function plus three point masses. | |
| May 29, 2012 at 10:28 | comment | added | kolik | I think that if f(x) = d mu(x) where mu(x) is a non-lattice distribution, then you can get some bounds < 1. I've seen results of this type in Esseen's (of Berry-Esseen fame) thesis. | |
| May 29, 2012 at 10:25 | comment | added | H A Helfgott | (If $f$ is a finite linear combination of delta functions (point masses), then one can just take derivatives and solve a polynomial equation - but of course this doesn't quite solve the general problem.) | |
| May 29, 2012 at 10:22 | history | asked | H A Helfgott | CC BY-SA 3.0 |