Timeline for Fourier Transform of compactly supported $L^1$ functions
Current License: CC BY-SA 3.0
3 events
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| Dec 5, 2013 at 19:39 | comment | added | User4891 | As I said, you don't get all holomorphic functions---there are also growth conditions which depend on the support of the function. The Payley-Wiener theorem gives a precise relationship between these and the support in the case of balls. It is possible to refine these for convex sets, using indicator functions. There are precise versions also for the case of distributions and smooth functions of compact support but the example of the FT for functions on the line suggests that there will be no such equivalences in the case of integrable functions. | |
| Dec 5, 2013 at 19:20 | comment | added | Nick S | But the Fourier Transform of any compactly supported tempered distribution is an entire function too. This tells us that the space of holomorphic functions is much bigger then the set I described. | |
| Dec 5, 2013 at 18:33 | history | answered | User4891 | CC BY-SA 3.0 |