Timeline for What is the 3-dimensional Fourier transform of $1/k^4$?
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| Dec 3, 2022 at 8:17 | comment | added | klempner | Two remarks which might be helpful. Firsly, you can easily reduce to the one-dimensional case by the usual methods (spherical coordinates). Secondly, for any $\alpha$, $|x|^\alpha$ has a natural interpretation as a distribution and has a Fourier transform (in the sense of a distributional parametrised integral) which is, up to a factor which depends on $\alpha$, |x|^{-1-\alpha}$. This was all worked out (using elementary methods without functional analysis in the 50‘s and 60‘s). I would be happy to supply references. | |
| Mar 21, 2022 at 10:28 | history | edited | YCor |
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| Mar 21, 2022 at 8:33 | comment | added | bathalf15320 | The good news is that the F.T. of a function of the above form is of the same nature. The best reference for you is probably the multi-volume "Generalised Functions" by Gelfand and Silov. | |
| Mar 21, 2022 at 8:30 | comment | added | bathalf15320 | Some remarks, hopefully helpful. By spherical symmetry, it süffifes to consider the one dimensional case, i.e., the functions $|x|^\alpha$ on the reals. In the classical framework, these present integrability problems either at $0$ or $\infty$ independent of the parameter. However they are always interpretable as distributions or generalised functions (not, with respect, a vague term but the standard soviet terminology), even tempered distributions and for these a coherent and elegant theory of Fourier transforms was developed by L. Schwartz right from the beginning. | |
| Mar 21, 2022 at 7:26 | answer | added | Carlo Beenakker | timeline score: 6 | |
| Mar 21, 2022 at 7:19 | comment | added | Denis Serre | I am puzzled with the reference which you give. The integrand behaves as $k^{-4}$ at the origin since the exponential is $\sim1$. This is definitely not integrable, even marginally (I should admit a discussion if it was $k^{-3}$) and there is no possible cancellation. At least the authors should explain what they mean by the very vague xpression "generalized function". | |
| Mar 21, 2022 at 7:09 | history | edited | Denis Serre | CC BY-SA 4.0 |
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| S Mar 21, 2022 at 5:16 | review | First questions | |||
| Mar 21, 2022 at 7:53 | |||||
| S Mar 21, 2022 at 5:16 | history | asked | HoangNguyen | CC BY-SA 4.0 |