Non-linear "Fourier analysis" Call a function of the following form a beep: $e^{-(\frac{x-\alpha}{\beta})^2}\sin(\rho x+\theta)$.  Given a real-valued function $f\in L^2(R)$ and a number $n$, I'm interested in the approximating $f$ as closely as possible by a linear combination of $n$ beeps.  
Does this particular type of non-linear regression problem have a literature?  
Do there exist good numerical techniques (perhaps after relaxing "best possible" in some controlled way) for solving this fitting problem?  
Improving an approximation sufficiently near the optimal one seems relatively straightforward, but first getting near the optimal approximation seems to involve some manner of combinatorial search.  Are there arguments from complexity theory that should dampen my expectations?
Are there theoretical results concerning how the error should vary with $n$ (perhaps with $f$ subject to some hypothesis, e.g. compact support or smoothness)?
Does the self-dual nature of the problem help in any way?  
Finally, I'm interested in anything I can learn along these lines, so feel free to tell me if you think I haven't quite asked the right question.
 A: Yes, http://en.wikipedia.org/wiki/Matching_pursuit#References
Yes, http://en.wikipedia.org/wiki/Tanh-sinh_quadrature
No, I don't think so.
Yes, http://en.wikipedia.org/wiki/Matching_pursuit#Properties
Yes, it gives an inner product.
No, I don't think so.
A: It may not help you, but I would recommend typing "quadratic Fourier analysis" into Google. That will give you links to a number of discussions of what is quite a big theme in additive combinatorics. However, the flavour of the problem on the real line is fairly different, so I don't think the results in additive combinatorics will directly answer the questions you have -- but they might just suggest one or two ideas.
A: Please search the literature for Chirplets. I remember reading articles published in IEEE journals 15+ years ago that describe the following method: Given a function/signal $f$, first calculate the STFT of $f$, then isolate the visible line segments on the graph of the spectrogram by treshholding. There are different ways to isolate line segments, e.g., apply Ridgelet transform or Radon transform to the spectrogram. Each line segment corresponds to a chirp. Roughly speaking, the linear combination of the chirps with the $n$-highest coefficients gives a (nonlinear) $n$-term approximation to $f$.
