Timeline for Exponential derivative of delta distribution?
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| May 31, 2021 at 18:05 | comment | added | reliquia | In this case, the range of the FT has to be larger than the space of tempered distributions, of course. As has already been mentioned in this thread, it is, in fact, a space of analytic functionals, i.e., elements of the dual of a suitable space of entire functions. Another canard is that there is something mysterious about a delta function with a complex argument. Despite it‘s name, it is simply a point measure and so can be defined as a Radon measure on any topological space, in particular as a distribution on any manifold. | |
| May 31, 2021 at 17:54 | comment | added | reliquia | This is the same old canard that one finds so often on MO. The fact that one can define the FT for tempered distributions doesn‘t mean that one cannot define it for more general ones. In the last 50, 60 years, a plethora of pairs of suitable spaces of test functions connected by the FT have been constructed and the latter then extends to the corresponding distribution spaces by duality. (The case of tempered distributions has a special symmetry due to the fact that the two spaces of test functions coincide). In particular, the FT has been extended to ALL distributions. | |
| Oct 26, 2017 at 18:50 | comment | added | Johannes Hahn | Does it? How so? Remember that compact support implies tempered. And the PWT for more general supports (say convex cones) still requires temperedness to make sense of the Fourier transform. | |
| Oct 26, 2017 at 17:08 | comment | added | Kagaratsch | What about the Schwartz's Paley–Wiener theorem? en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem It seems to suggest that tempered distribution is not the only admissible kind. | |
| Oct 26, 2017 at 16:57 | history | answered | Johannes Hahn | CC BY-SA 3.0 |