Knowlpedia

Lemma

Orthogonality of periodic Fourier characters

Distinct integer-frequency characters have zero averaged Hermitian product.

Statement

For on a unit cell,

∫[0,1]dem(x)en(x)‾ dx={1,m=n,0,m≠n.\int_{[0,1]^d}e_m(x)\overline{e_n(x)}\,dx =\begin{cases}1,&m=n,\\0,&m\ne n.\end{cases}

Thus they are orthonormal for the normalized L2L^2 inner product.

Proof

The integrand is em−ne_{m-n}. Fubini factors the integral into one-dimensional integrals. For an integer k≠0k\ne0, ∫01e2πiks ds=(e2πik−1)/(2πik)=0\int_0^1e^{2\pi iks}\,ds=(e^{2\pi ik}-1)/(2\pi ik)=0; for k=0k=0 it equals one.

Products without conjugation

Likewise ∫emen dx\int e_me_n\,dx equals one precisely when m+n=0m+n=0, and zero otherwise. Keeping track of the conjugation is essential when computing a real wave's quadratic average.

References