Say I have 10,000 data in 2-D and I want to fit a curve to them. There are many functional forms this curve could take -- polynomial, B-spline, trigonometric, and so on. I've decided that I only want to use 4 parameters.
Is there a way to figure out what is the most accurate functional form? That is, considering all possible functions with 4 parameters, which one fits the best in, e.g., an $L_2$ sense?
edit: maybe I should ask about the most accurate function with the same V-C dimension as a polynomial of degree 4 rather than "4 parameters"?