Timeline for Interpret Fourier transform as limit of Fourier series
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Jan 16, 2016 at 3:10 | answer | added | Nik Weaver | timeline score: 2 | |
| Jan 15, 2016 at 5:43 | history | edited | Lao-tzu | CC BY-SA 3.0 |
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| Jan 14, 2016 at 16:25 | comment | added | Qiaochu Yuan | @Lao-tzu: by $A^{\vee}$ i mean the Pontryagin dual of $A$. By $\widehat{\mathbb{Z}}$ I mean the profinite integers. | |
| Jan 14, 2016 at 15:27 | comment | added | Yemon Choi | @Dirk But I think the question is hoping to see the continuous FT as a limit of the discrete one, and I think that perspective while obviously attractive can be misleading if applied without caution | |
| Jan 14, 2016 at 15:23 | comment | added | Yemon Choi | The Bohr compactification of ${\mathbb R}$ looks nothing like a torus... | |
| Jan 14, 2016 at 11:31 | comment | added | Dirk | Interpreting Fourier inversion as a limit is not only interpreting but is the right way to see Fourier inversion in $L^2$. One defines the Fourier transform by extending it from Schwartz space (or $L^1\cap L^2$) to $L^2$ and similar for the inverse. To be concrete one can use the limit $\lim_{T\to\infty} \int_{-T}^T \hat f(\xi) \exp(i x\xi) d\xi$… | |
| Jan 14, 2016 at 9:31 | history | edited | Lao-tzu | CC BY-SA 3.0 |
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| Jan 14, 2016 at 8:14 | history | edited | Lao-tzu | CC BY-SA 3.0 |
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| Jan 14, 2016 at 8:06 | history | edited | Lao-tzu | CC BY-SA 3.0 |
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| Jan 14, 2016 at 7:57 | answer | added | Watson Ladd | timeline score: -1 | |
| Jan 14, 2016 at 7:03 | history | edited | Lao-tzu | CC BY-SA 3.0 |
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| Jan 14, 2016 at 6:42 | comment | added | Lao-tzu | What do you mean by $\mathbb{Z}^ˆ∨$? | |
| Jan 14, 2016 at 6:33 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 3.0 |
"series" and "transform" had to change places
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| Jan 14, 2016 at 6:31 | history | edited | Lao-tzu | CC BY-SA 3.0 |
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| Jan 14, 2016 at 6:24 | history | edited | Lao-tzu | CC BY-SA 3.0 |
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| Jan 14, 2016 at 6:23 | comment | added | Qiaochu Yuan | A simpler and related question is to interpret the "limit" of the Pontryagin duality relationships $\mathbb{Z}_n^{\vee} \cong \mathbb{Z}_n$ as the Pontryagin duality relationship $\mathbb{Z}^{\vee} \cong S^1$. The closest I know how to get is to take categorical limits / colimits, which gets you the duality $\widehat{\mathbb{Z}}^{\vee} \cong \mathbb{Q}/\mathbb{Z}$. | |
| Jan 14, 2016 at 6:15 | history | asked | Lao-tzu | CC BY-SA 3.0 |