Projection onto zero-mean functions

For $0<\mu(X)<\infty$ , the zero-mean projection is

P_0f=f-\frac{1}{\mu(X)}\int_X f\,d\mu.

It is linear, its output has zero mean, and $P_0^2=P_0$ . It removes exactly the constant component.

Norms and parameters

On $L^2(X,\mu)$ , it is the orthogonal projection onto the complement of the constants, and $\|P_0f\|_2\le\|f\|_2$ . For $1\le p\le\infty$ , Hölder's inequality gives $\|P_0f\|_p\le2\|f\|_p$ . The notation must specify $X$ and $\mu$ ; averaging only an auxiliary variable leaves the remaining variables as parameters.