The following question arose. What functions do they satisfy? $\widehat{f}=f$ with $f\in L^2(\mathbb{R})$? Only the function $f=0$?
Asked
Active
Viewed 41 times
0
-
1gaussian too. explicitly which i believe depends on your convention for the FT – Calvin Khor May 26 '22 at 05:19
-
$\widehat{f}(x):=\int \mathrm{e}^{-ix\cdot \xi} f(\xi)d\xi$ – eraldcoil May 26 '22 at 05:20
-
and also there should be some of the form poly * that gaussian, since poly turns to derivatives which then become poly again. Going to guess hermite polys – Calvin Khor May 26 '22 at 05:22
-
Is the following valid? $f=\check{f} \Rightarrow \widehat{f}=\check{f} \Rightarrow \widehat{(\widehat{f})}=f\Rightarrow f(-x)=f(x)$? – eraldcoil May 26 '22 at 05:22
-
2Hermite functions of degree a multiple of 4. Google "hermite functions as eigenvectors of fourier transform." – Stephen Montgomery-Smith May 26 '22 at 05:39
-
There are many functions that are fixed points of the Fourier transform. See https://mathoverflow.net/questions/12045/what-are-fixed-points-of-the-fourier-transform – Stefan Lafon May 26 '22 at 09:52
-
Also see this https://math.stackexchange.com/questions/118078/which-functions-fourier-transform-is-the-function-itself – Stefan Lafon May 26 '22 at 10:06