Zero-mean function - Knowlpedia

On a measure space $(X,\mu)$ with $0<\mu(X)<\infty$ , an integrable scalar or finite-dimensional vector-valued function $f$ is zero mean if

\int_X f\,d\mu=0.

Equivalently, its normalized average $\mu(X)^{-1}\int_X f\,d\mu$ vanishes. Vector integrals are interpreted componentwise.

Variables are part of the definition

A family $f(x,y)$ can have zero mean in $y$ for each fixed $x$ , without being independent of $x$ . On a periodic cell, zero mean is equivalent to vanishing of the zero Fourier coefficient. Changing the measure or the averaging variables can change whether the condition holds.