Let $A: L^2(R^n)\to L^2(R^n)$ be a non-linear operator. Is it known when it's possible to split $A$ into a composition of a linear operator $B: L^2(R^n)\to (L^2(R^n))^k$ and a local operator $C: (L^2(R^n))^k\to L^2(R^n)$? Are there any techniques to do so?
Edit:
Perhaps I should explain what I am really looking for and why. The question arose from the need to efficiently compute certain functions in $R^2\to R$, based on the input $R^2\to R$ and certain operators. There are basically two approaches to do that in the industry: if the operator is local or has very small support radius then compute locally; if the operator is linear then do FFT (Fourier transform), compute locally, and do inverse FFT.
The question is basically about operators that can be split into a composition something local and spacial domain and something local in the frequency domain.
Sometimes operators are not local at all and not quite linear either, but it is still possible to compute them efficiently. A typical example: compute the slope of the signal given by transmission matrix $f: R^2\to R$ and a linear operator $K: L^2(R^2)\to L^2(R^2)$. Let $g=K\star f$. Although $f\to g$ is linear, and $f\to grad(g)$ is linear, $f\to slope(g)$ is not.
However, non-linearity of the slope never stopped anybody from computing it efficiently: one just perform the computation in two steps:
Step 1: compute $grad(g)$ using FFT: $grad_K: L^2(R^2)\to (L^2(R^2))^2: f\to grad(K\star f)$
Step 2: compute $slope_K(g): (L^2(R^2))^2\to L^2(R^2)$ locally from $grad_K(g)$.
Therefore the question: how far one can extend the above approach? What operators can be split into compositions of a local and a linear one? (Possibly) more generally, what operators can be split into a finite sequence of compositions of local and linear operators?