Consider a noisy time series y_i and that I have waited until end of experiment. I now want to fit a nonlinear parameterized function F(A,d,k; t) to determine A,d,k.
So I can happily Newton-Raphson my way through | y - F|^2. But gosh the y's are noisy. I can't stop myself from filtering them first to throw away some high frequency junk. I think it helps. So I am minimizing |L y - F|^2 ... All is good. Except. To design L, I did some handwaving about what I thought would be good passband, ripple, etc. But who cares? What I really want is for |L y - F|^2 to converge quickly and for A,d,k to be "most" representative of the data.
This takes me down the road that I should be thinking of filters with L^t F = 0 so that filter doesnt change A,d,k.... It also makes me think that L could profitably help as a preconditioner. This seems like a different design optimization.
I'm also thinking it is nothing new and probably written up in textbooks everywhere. Any pointers to them would be appreciated, thanks.