Knowlpedia

Definition

Zero Fourier mode

The constant component of a periodic Fourier expansion, equal to its normalized mean.

The zero Fourier mode of an integrable periodic function is the constant function f^(0)e0=f^(0)\widehat f(0)e_0=\widehat f(0). On a unit cell,

f^(0)=∫[0,1]df(x) dx.\widehat f(0)=\int_{[0,1]^d}f(x)\,dx.

Thus the zero is the normalized mean.

Partial averaging

For f(x,y)f(x,y), taking the zero mode only in yy leaves a function of xx. Saying that a coefficient has zero frequency in one angular variable does not imply independence of every auxiliary variable.