Timeline for Fourier series of $e^{(\cos(\pi x) - m)^2}$
Current License: CC BY-SA 4.0
12 events
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| Sep 14, 2022 at 16:57 | comment | added | gaspardb | The term "periodic Gaussian" is entirely mine and I wrote it without thinking. But thanks a lot for your comment, because "your" periodic Gaussian seems better suited to my application (I don't mind having troubles evaluating $f(x)$ if I can evaluate $c_j$ easily). You gave me a lot of interesting things to read! Besides, the relation between "my" PG and "yours" does not seem straightforward. | |
| Sep 14, 2022 at 11:10 | comment | added | Gro-Tsen | Regarding terminology: Is the term “periodic Gaussian” yours? Because what I would call a periodic Gaussian is (a shift of) the function whose Fourier coefficients are $c_j := \exp(-j^2)$ (up to scale), or equivalently a sum of periodically translated Gaussians: this of course should look somewhat like yours, but seems like a more natural object since it is, e.g., Green's function for the heat equation on the circle (which is how I define a Gaussian on Riemannian manifolds in general). | |
| Sep 14, 2022 at 10:06 | history | edited | gaspardb | CC BY-SA 4.0 |
$x$ is not in the formula
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| Sep 13, 2022 at 15:37 | history | edited | gaspardb | CC BY-SA 4.0 |
added 124 characters in body
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| Sep 13, 2022 at 11:50 | history | edited | gaspardb | CC BY-SA 4.0 |
added 287 characters in body
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| Sep 13, 2022 at 11:46 | comment | added | gaspardb | My take will be to truncate the sum, but I need to find efficient ways to do so -- I also need to read about hypergeometric series. I was wondering if something simpler existed! | |
| Sep 13, 2022 at 11:45 | comment | added | gaspardb | Indeed I was hoping to have the very neat formula of the post you linked (which I added to my question). Unfortunately I did not know how to deal with $\cos 2 \pi x + \cos \pi x$. Could you elaborate on the "summation formula" to deal with that? | |
| Sep 12, 2022 at 21:40 | answer | added | Carlo Beenakker | timeline score: 4 | |
| Sep 12, 2022 at 21:18 | comment | added | Thomas Kojar | the sum you wrote, assuming correct, should be ok numerically if you truncate it at some point, right? | |
| Sep 12, 2022 at 21:17 | comment | added | Thomas Kojar | how about first turning the squared cosine into cos(2x) and then maybe using summation formula to turn into to just cosine? Which has a neat formula mathoverflow.net/questions/272505/… | |
| S Sep 12, 2022 at 20:51 | review | First questions | |||
| Sep 13, 2022 at 6:02 | |||||
| S Sep 12, 2022 at 20:51 | history | asked | gaspardb | CC BY-SA 4.0 |