Timeline for $L^2$ norm of self-convolution
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 11, 2023 at 17:00 | comment | added | Alek Westover | Thanks! I'll think about it. I like the supp f constraint | |
| Aug 11, 2023 at 15:49 | comment | added | Pietro Majer | Sorry, I understood you wanted a bound $\|g\|\le c\|f\|^2$, for which $c=\sqrt n$ is trivially optimal. As $\|f\|$ increases, $f$ has to be more and more close to a face of the simplex $\{f: f_i\ge0,\sum_if_i=1\}$, that is, more and more coordinates of $f$ are small. So I guess it may be useful to consider the optimization problem $\max_f \|f*f\|$ with an additional constraint $\text{supp} f=J\subset\{1,\dots,n\}$. Note that the limit case $\|f\|=1$ implies $f=e_i$ for some $i$ and $\|f*f\|=1$. | |
| Aug 11, 2023 at 12:12 | comment | added | Alek Westover | This seems true, but very weak if f has large L2 norm say larger than n^{1/4}. Can anything be said in that regime? My intuition is that g should have smaller L2 norm than f and usually much smaller. Are there counter examples? | |
| Aug 11, 2023 at 12:04 | vote | accept | Alek Westover | ||
| Aug 11, 2023 at 12:11 | |||||
| Aug 11, 2023 at 10:38 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| Aug 11, 2023 at 6:27 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| Aug 11, 2023 at 6:11 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| Aug 11, 2023 at 6:00 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| Aug 11, 2023 at 5:53 | history | answered | Pietro Majer | CC BY-SA 4.0 |