Yes, I have missed the point.
Note that Vadim said he wanted to find f(). You're assuming f() is known.
The model is very strange (to me, at least). It's not obvious to me that
it's even identifiable. (Sorry that I don't have anything constructive to
add.)
Andy
But if you have a grid in (x_1,x_2) plane and y[i,j] values in the nodes (or you can interpolate an irregularly spaced data) then you may solve your problem with Fourier transform, get fourier coefficients and find your function (i.e tabulated values) with inverse fourier transform.
Precisely it would look like that
Suppose there is a square grid in plane. Your model is y[i,j]=a0+a*(z<-cbind(c(f1,f2,f3...fn), c(f1,f2,f3,...fn)....))+t(z), i,j<N (because function f(x) is a table then may painlessly suppose that a2 is included in it). Then you convolve row-wise and column-wise y[i,j] with sin(2*pi*w*i/N), cos(...) (w<N). "Best-fit" Fourier coefficient (with multiplier "a" for columnwise convolution) is an average of vector you got after convolution. a0 may be found then convolving with 1 (i.e. just calculate average). After you got all coefficients "a" may be found with linear fit and (fi) with inverse fourier transform.
Fourier transform is little tricky when dealing with partial sums (with truncated series) because a result is not always smooth although in theory it should be, but why not give it a try.
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