Timeline for Integrability of a function under a condition on its Fourier transform
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 19, 2020 at 21:03 | comment | added | Paul | thank's so much | |
| Jun 19, 2020 at 12:56 | comment | added | Giorgio Metafune | If $\phi_R(x)=\phi \chi_{(0,R)}$, then $\phi_R \to \phi$ and $\hat{\phi_R} \to \hat{\phi}$ in $L^2$. But $\hat{\phi_R}=\int_0^R \phi (x) \sin(\xi x)dx \to \psi$ pointwise. | |
| Jun 19, 2020 at 11:46 | comment | added | Paul | @ Giorgio Metafune According to my teacher, we can only express a fourier transform or its inverse by an integral if the function is in L ^ 1, here the function is in L ^ 2, the inversion formula $\psi (\xi)=\int_0^\infty \phi(x)e^{(i\xi x)}\, dx.=2 \int_a^\infty \phi(x)\sin (\xi x)\, dx.$ is not justified | |
| Jun 17, 2020 at 18:59 | comment | added | Paul | @ Giorgio Metafune Thank's so much | |
| Jun 17, 2020 at 16:15 | history | answered | Giorgio Metafune | CC BY-SA 4.0 |