fourier basis

definition

• let $\omega =e^{-2\pi i/n},({\dot{1}}^2=-1)$
• define vectors in $R^n$: $u^{(j)}(t)=\frac{1}{\sqrt{n}}{\omega }^{tj},t=1,\dots ,n.$

orthogonality of fourier basis

• fourier vectors all have unit norm: $\Vert u^{(j)}{\Vert }_2=1,j=1,\dots ,n$
• they are mutually orthogonal: $(u^{(j)}{)}^H{u}^{(i)}=0,\forall i\ne j$