Timeline for Conditions for absolute continuity in the Bochner-Schwartz theorem
Current License: CC BY-SA 4.0
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| May 25, 2020 at 19:15 | comment | added | user78370 | @LiviuNicolaescu Indeed, but here I'm hoping to capture a more general phenomenon. For example, both the delta Dirac distribution and the Riesz kernel $f(x)=|x|^{n-\alpha}$ have absolutely continuous Fourier transforms, but neither are in $L^2$. Both of these distributions happen to have some form of decay (much weaker than what would be imposed by continuity+$L^2$, for example), which is what I suspect has something to do with absolute continuity. | |
| May 25, 2020 at 18:44 | comment | added | Liviu Nicolaescu | The regularity of $g$ is easier It is related to the decay properties of the Fourier transform. If $f$ is in $L^2$ then $\mu$ is absolutely continuous with $g\in L^2(\mathbb{R}^n, dx)$. | |
| May 25, 2020 at 10:13 | history | asked | user78370 | CC BY-SA 4.0 |