Orthogonality in $L^2$ of functions such that their Fourier transform are supported in disjoint sets

Question

If the functions $f_j$ defined on $\mathbb{R}^n$ have Fourier transforms $\hat{f}_j$ supported in disjoint sets, then they are orthogonal in the sense that $$\|\sum_j f_j\|_{L^2}^2=\sum_j\|f_j\|_{L^2}^2.$$ How can I prove it in detailed manner?

Answer 1

They're not orthogonal is the sense you have written. They are orthogonal in the $L^2$ Hilbert space sense because the Plancherel Theorem (a.k.a. Parseval) gives $$ \|\sum_j f_j \|^2=\|\sum_j \hat{f_j}\|^2=\sum_j\|\hat{f_j}\|^2=\sum_j \|f_j\|^2. $$ Mutual $L^2$-orthogonality is preserved under the Fourier transform because it is unitary on $L^2$.