Timeline for Fourier transform of (real) exponential
Current License: CC BY-SA 2.5
5 events
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| Mar 24 at 19:34 | comment | added | Zeekless | Is there a reason why the Schwartz space is more widely acknowledged and popular? Do we lose something when moving to smaller sets of test functions (that are closed under Fourier transforms)? Having a Fourier transform of a real exponential seems like a significant achievement to me, so I do not understand why most sources do not extend the class of distributions beyond the tempered ones, unless there is a good reason not to do so. | |
| Oct 13, 2010 at 17:20 | comment | added | B R | To elaborate on why smooth compactly supported test functions fail: Using asymptotic methods, the Fourier transform of the (smooth compactly supported) function $\phi_a(x)=e^{-1/(a^2-x^2)}$ (and $\phi(x)=0$ for $|x|\ge a$) has exponential decay depending on $a$, so that for any exponential function $e^{\lambda x}$, we can find a $\phi_a$ so that $\int e^{\lambda x}\hat\phi_a(x)$ diverges. Thus the distributional Fourier transform of $e^{\lambda x}$ does not exist (because we can't integrate it against the Fourier transform of all test functions). | |
| Oct 13, 2010 at 15:43 | history | edited | Richard Borcherds | CC BY-SA 2.5 |
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| Oct 13, 2010 at 14:45 | comment | added | Willie Wong | The spaces you describe are, I think, the Gelfand-Shilov spaces $S^a_a$. For a Fourier characterization of them see jstor.org/stable/2161498 . I don't know much about them personally (having only skimmed a bit of literature); they are not particularly useful to me because they are either analytic or quasi-analytic (I forgot which), with the notable property that the test function space does not contain any function of compact support (in fact, I think they may not even have functions which vanish on open sets, as long as some condition on the index $a$ is obeyed.) | |
| Oct 13, 2010 at 13:50 | history | answered | Richard Borcherds | CC BY-SA 2.5 |