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.
Best Answer
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.