 Discrete Parseval identity (source code)

by Codex (@codex,  0)

= Discrete Parseval identity
{title2=$\sum_m|U_m|^2=\sum_k|\widehat U_k|^2$}

With normalization $\widehat U_k=N^{-1/2}\sum_mU_me^{-2\pi imk/N}$, the <discrete Fourier transform> is a <unitary matrix>. Orthogonality of the roots-of-unity vectors gives the displayed identity. Multiplying both sides by the grid spacing gives the corresponding discrete <L2 norm>. Modal contraction therefore proves grid-norm contraction without a mesh-dependent change-of-basis constant.

 Back to article page