Given $f\in L^2(\mathbb{R}^n)$. For any $\epsilon \in(0,x)$, define $$\hat{f_\epsilon(\xi)}=\chi_{[x-\epsilon,x+\epsilon]}(|\xi|).$$ Where $\hat{f_\epsilon(\xi)}$ is the Fourier Transform $f$.
I am wondering how to compute the following:
- $f$
- $\|f\|$
- $\|\hat{f}\|$
- $\|\chi_{[x-\epsilon,x+\epsilon]}(|\xi|)\|$.
I know that \begin{eqnarray*} % \nonumber to remove numbering (before each equation) f(x) &=& 1/(2\pi)^n\int \hat{f_\epsilon(|\xi|)}e^{ix\xi}d\xi \\ &=&1/(2\pi)^n\int \chi_{[x-\epsilon,x+\epsilon]}(|\xi|)e^{ix\xi}d\xi$\\ &=&1/(2\pi)^n\int_{x-\epsilon}^{x+\epsilon}e^{ix\xi}d\xi\\ &=&1/(2\pi)^n\cdot 1/ix\cdot (e^{ix(x+\epsilon)}-e^{ix(x-\epsilon)}). \end{eqnarray*}
But this is not leading me to anything interesting!!
I also think that after obtaining $f$ I can go on and use $\|f\|_{L^2}=1/(2\pi)^{n/2}\|\hat{f}\|_{L^2}$ to obtain $\hat{f}$. I need some help please.