Is it possible to write a sinus function as a linear combination of different frequency sinus?

Question

I'm solving a engeering problem that, after several steps and simplifications, lead to the following mathematical problem:

Is it possible to find a1 , a2, a3, an... which satisfies the following expression?

a1*sin(w1*t) + a2*sin(w2*t) + a3*sin(w3*t) +...+ ansin(wnt) ≅ sin(w*t)

Considering that w1, w2, w3, w4 > 100*w .

Or maybe an approximation?

Answer 1

In exact form, that's impossible. The linear combination of sinusoids of different frequencies does not produces "new" frequencies - you'd need some non-linear operations for that. Read about Fourier analysis.

Regarding approximations... that would highly depend on how you measure the goodness of the approximation. If using the mean squared deviation (over the full real line) then, again, Fourier analysis shows that no combination gives a better approximation than the null combination ($a_1 = a_2 = \cdots =0$).