Effect of repeated subtraction of the average of average function values in coordinate directions 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?

 A: 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:
$$f^*(x,y)=f(x,y)-{1\over|I|}\int_If(s,y)ds-{1\over|J|}\int_Jf(x,t)dt+{1\over|I\times J|}\int_{I\times J}f(s,t)dsdt$$
