**Questions:**


assuming


$$a\lt b,\ c\lt d;\ \ (x,y)\in [a,b]\times[c,d];\ \ f\_0: (x,y)\mapsto z\in\mathbb{R};\ \ |a|,\ |b|,\ |c|,\ |d|,\ |z|\lt\infty$$  

$$0\quad\lt\quad\left|\int\_a^b{f\_0(x,y)dx}\right|,\ \ \left|\int\_c^d{f\_0(x,y)dy}\right|$$
$$f\_{i+1}(x,y) := f\_i(x,y)-\frac{(d-c)\int\_a^b{f\_i(t,y)dt}\ +\ (b-a)\int\_c^d{f\_i(x,t)dt}}{2(b-a)(d-c)}$$



> 
> **what is known** about
> 
> 
> * the convergence of the iteration defined above?
> * provided convergence, non-trivial properties of $$f^\*(x,y):=\lim\_{i\to \infty}{f\_i(x,y)}$$
> * the evaluation of $f^\*(x,y)$, i.e. the efficient calculation of its values?
> 
> 
> 



Effect of repeated subtraction of the average of average function values in coordinate directions
Denote $I:=[a,b]$, $J:=[c,d]$ and say $f$ is in $L\_2(I\times J)$. You are iterating the bounded linear operator ${\bf1}-{1\over2}(P+Q)$, where $Pf(x,y):={1\over|I|}\int\_If(s,y)ds$ and $Qf(x,y):={1\over|J|}\int\_Jf(x,t)dt$ take the mean of $f$ on the first, resp. second variable; that is, $P$ and $Q$ are the linear (ortho)projectors on the closed subspaces $L\_2(J)\subset L\_2(I\times J)$ resp., $L\_2(I)\subset L\_2(I\times J)$ of the functions which are constant wrto the first, resp. second variable. So $P^2=P$, $Q^2=Q$ and $PQ=QP$ by Fubini's theorem. For commuting projectors, $(P+Q)^m= (2^m-2)PQ+P+Q$; since ${\bf1}-P$ and ${\bf1}-Q$ are themselves a pair of commuting projectors this gives
$$\begin{align}\Big[{\bf1}-{1\over2}(P+Q)\Big]^m&={1\over2^m}\Big[({\bf1}-P)+{(\bf1}-Q)\Big]^m=\\&=\Big(1-{1\over2^{m-1}}\Big)({\bf1}-P)({\bf1}-Q)+{1\over2^{m}}({\bf 1}-P+{\bf 1}-Q),\end{align}$$
that converges in operator norm to $({\bf1}-P)({\bf1}-Q)$, the orthoprojector onto $\ker P\cap\ker Q$, the space of functions with $\int\_If(s,y)ds=\int\_Jf(x,t)dt=0$ for (a.e.) $(x,y)\in I\times J$. Everything (but the Hilbert language) holds true in the space $L\_1(I\times J)$ too. In the original notation:
$$f^\*(x,y)=f(x,y)-{1\over|I|}\int\_If(s,y)ds-{1\over|J|}\int\_Jf(x,t)dt+{1\over|I\times J|}\int\_{I\times J}f(s,t)dsdt$$


