Timeline for Is there a Fourier Analytic way to approximate volume?
Current License: CC BY-SA 3.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Oct 23, 2017 at 1:19 | answer | added | Igor Rivin | timeline score: 1 | |
| Oct 23, 2017 at 1:12 | comment | added | Anton Fetisov | In principle it can be computed and gives an explicit complicated formula via all $\mathbf k$'s, but in fact it involves calculating all faces, edges, vertices etc, so it's more complex than the initial problem. It doesn't look like one could evaluate an integral of n-dimensional delta functions without computing intersections, but of course it's not a proof that some algorithm couldn't exist. | |
| Oct 23, 2017 at 1:09 | comment | added | Anton Fetisov | It's certainly possible to calculate the volume via Fourier analysis: $Vol(A)=\int 1_A dx$, where $1_A$ is the indicator function of $A$ equal to the product of $\Theta(\mathbf k \mathbf x -1)$ over all vectors $\mathbf k$ defining the linear inequalities. Here $\Theta$ is Heaviside's function. Since $\int 1_A dx = (\int 1_A e^{i p x} dx)|_{p=0}$ and the Fourier transform of a product is a convolution of Fourier transforms, we get a Fourier calculation for volume. The F.t. of each $\Theta$ is a sum of delta-functions on the space and on orthogonal to $\mathbf k$'s. | |
| Oct 22, 2017 at 21:45 | history | edited | Turbo | CC BY-SA 3.0 |
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| Oct 22, 2017 at 21:37 | history | edited | Turbo | CC BY-SA 3.0 |
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| Oct 22, 2017 at 21:32 | history | undeleted | user106049 | ||
| Oct 22, 2017 at 21:29 | history | deleted | user106049 | via Vote | |
| Oct 22, 2017 at 21:28 | history | asked | Turbo | CC BY-SA 3.0 |