Experimental comparison of G-decomposition of waves vs. Fourier

Here F is the Fourier style decomposition, for 5 points , and G is the new decomposition I've been working with, written also for five points:

So I tried not to cherry pick these values, I've gotten similar results with every set of points I've tried so far, but let's let the points be [1,1],[3,2],[5,5],[7,3],[9,2] and solve for the coefficients:

I think the problem with the Fourier is that it maxes to 6 height when the largest point is only 5, the G decomposition seems more reasonable,



**Note**
I don't know if the exponents in G cause the coefficients to get too large with a large number of points that will require some more investigation...And of course using powers of trigonometric functions might make them too costly to computer, but there might be some use cases where that isn't as important as the best fit to the points... There might even be some variations between these two where you use some small powers of sines and cosines and some multiples of frequencies...