Sinusoidal fitting when frequencies are known however some samples are missing? I'm working in electronics and signal processing field. I have a question regarding data fitting  and I appreciate any help.
Suppose we have a signal composed of three sines with frequencies of $f=[10,20,45]$, i.e $$x=A_1\sin(w_1t+\phi_1)+A_2\sin(w_2t+\phi_2)+A_3\sin(w_3t+\phi_3)$$
$$=A_1\sin(2\pi*10t+\phi_1)+A_2\sin(2\pi*20t+\phi_2)+A_3\sin(2\pi*45t+\phi_3)$$
Suppose I want to  fit some sines to the data sampled with sample rate of half of Nyquist rate from the above signal (i.e. instead of sampling with frequency of $90$ I've sampled with frequency of $45$). And somehow, I know frequency (and maybe phase) parameters and I only want to estimate amplitudes? Is it possible? If it is, how? Note that here I have not observed Nyquist rate,
I have seen similar thing in as Non-uniform random sampling in Sparse Recovery literature, would you please explain the algorithm.
 A: $$x=A_1\sin(w_1t+\phi_1)+A_2\sin(w_2t+\phi_2)+A_3\sin(w_3t+\phi_3)$$
$w_1$ , $w_2$ , $w_3$ are known. The unknown are $A_1$ , $A_2$ , $A_3$ , $\phi_1$ , $\phi_2$ , $\phi_3$ .
With : $\begin{cases}
B_1= A_1\sin(\phi_1) \quad ;\quad  C_1= A_1\cos(\phi_1)\\
B_2= A_2\sin(\phi_2) \quad ;\quad  C_2=A_2\cos(\phi_2)\\
B_1= A_3\sin(\phi_3) \quad ;\quad  C_3= A_3\cos(\phi_3)
\end{cases}\quad$  rewrite the equation as :
$$x=B_1\cos(w_1t)+C_1\sin(w_1t)+B_2\cos(w_2t)+C_2\sin(w_2t)+B_3\cos(w_3t)+C_3\sin(w_3t)$$
This is a linear equation relatively to $B_1$ , $C_1$ , $B_2$ , $C_2$ , $B_3$ , $C_3$ 
You can perform a linear regression (least squares fitting) which will give the approximates of $B_1$ , $C_1$ , $B_2$ , $C_2$ , $B_3$ , $C_3$ . Note : "Linear" because the equation is linear relatively to those six unknowns.
Finally, compute :
$$\begin{cases}
A_1= \sqrt{B_1^2+C_1^2} \quad ;\quad  \phi_1= \tan^{-1}\left(\frac{B_1}{C_1}\right)\\
A_2= \sqrt{B_2^2+C_2^2} \quad ;\quad  \phi_2= \tan^{-1}\left(\frac{B_2}{C_2}\right)\\
A_3= \sqrt{B_3^2+C_3^2} \quad ;\quad  \phi_3= \tan^{-1}\left(\frac{B_3}{C_3}\right)
\end{cases}\quad$$
Doesn't matter if some samples are missing and if the times of sampling are not regularly distributed.
it is very important for the reliability and the accuracy that $w_1$ , $w_2$ , $w_3$ be exactly known (at least with high accuracy).
