I have been trying for several months to solve what looks (on the surface) like a simple problem. I have 9 data points (t,y) and it is known that these points represent the turning points of a continuous waveform which is the sum of three sinusoids:
$y(t) = a.cos(u.t) + b.sin(u.t) + c.cos(v.t) + d.sin(v.t) + e.cos(w.t) + f.sin(w.t)$
So we know that $y(t)$ passes through the nine points and we also know that the derivative $dy(t)/dt$ is zero at the nine points (see example graph at the end of this post).
So we have 18 equations in nine unknowns (a,b,c,d,e,f,u,v,w), nine from $y(t)$ and nine from $dy(t)$.
I have tried a number of approaches, especially linear algebra (e.g least squares, SVD, etc) but am stuck at every turn! It is easy to find an expression for (a,b,c,d,e,f) in terms of (u,v,w), thus reducing the size of the problem from 9 variables to 3, but after that the problem is fighting me!
Can anyone help?
Many thanks!
In this image, the nine blue circles are the known data points. The smooth red curve is what I am seeking:-
