Timeline for Interpret Fourier transform as limit of Fourier series
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 15, 2016 at 17:09 | comment | added | Watson Ladd | I fixed that in an edit. Thanks for pointing that out | |
| Jan 15, 2016 at 17:08 | history | edited | Watson Ladd | CC BY-SA 3.0 |
added 3 characters in body
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| Jan 15, 2016 at 12:26 | comment | added | Jean Duchon | @Watson Ladd ... but your sentence "it is completely determined by a finite number of Fourier coefficients" is wrong, whence the downvotes to your answer, by people (not me) who may well know the basics of Fourier analysis. The Nyquist sampling theorem, or some form thereof, is certainly part of that "standard material". | |
| Jan 15, 2016 at 1:55 | comment | added | Watson Ladd | A bandlimited function is completely determined by its samples if the sampling rate is twice the maximum frequency. That's the Nyquist sampling theorem. The result of a Fourier transform has infinitely many frequency components where the frequencies go down to zero: it's time limited samples where the number of frequency components is finite. This is all standard material in signal processing textbooks, which I cited by the name of the theorem. | |
| Jan 14, 2016 at 16:03 | comment | added | Dirk | If the Fourier transform of f has compact support, then f is not compactly supported (follows from Heisenberg uncertainty principle, or, more nicely from the uncertainty principle by Donoho and Stark). Hence, there are countable many nonzero samples that determine f completely. If bandlimited functions could be compactly supported, signal processing would be considerably much easier. | |
| Jan 14, 2016 at 15:23 | comment | added | Yemon Choi | Moreover, what does your last paragraph actually mean? It seems unfortunately close to hand-waving | |
| Jan 14, 2016 at 14:31 | comment | added | Jean Duchon | How can elements of an infinite dimensional space (that of functions whose Fourier transforms have supports contained in $[-R,R]$) be "completely determined" by finitely many coefficients ?? | |
| Jan 14, 2016 at 7:57 | history | answered | Watson Ladd | CC BY-SA 3.0 |