I'm curious to understand how to compute the fourier transform for the following characteristic function $\chi_{[-1,1]^n}$?
I have only been able to get this far:
$\mathcal{F}\chi_{[-1,1]^n}(\xi)=\int_{-\infty}^{\infty}\chi_{[-1,1]^{n}}(x)e^{-2\pi i\xi x}dx$.
How do i proceed further? A bit confused on how the limits of the integral will change?
Thanks for the help! Really appreciate it!
Actually your formula $$\mathcal{F}\chi_{[-1,1]^n}(\xi)=\int_{-\infty}^{\infty}\chi_{[-1,1]^{n}}(x)e^{-2\pi i\xi x}dx$$should be $$\mathcal{F}\chi_{[-1,1]^n}(\xi)=\int_{-\infty}^{\infty}\chi_{[-1,1]^{n}}(x)e^{-2\pi i\xi\cdot x}dx$$(in your version you have the exponential of a vector.) Noting that $$e^{-2\pi i\xi\cdot x}=\prod_{k=1}^ne^{-2\pi i \xi_kx_k}$$you see the whole thing reduces to $$\int_{-1}^1\dots\int_{-1}^1\prod_{k=1}^ne^{-2\pi i \xi_kx_k}\,dx_1\dots dx_n;$$you can find that integral by just calculus.