hello community.
I need some help
given $k = \sum A_q e^{ia_q s}$ where $k$, and $s$ is known. Can $A_q$ s now be expressed in terms of $a_q$ .
any help in that direction will be appreciated.
hello community. I need some help given $k = \sum A_q e^{ia_q s}$ where $k$, and $s$ is known. Can $A_q$ s now be expressed in terms of $a_q$ . any help in that direction will be appreciated. 


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No, you cannot, at least not at this level of generality. Consider for example $a_q = \frac{\pi}{q}$, $s=1$ and $k=0$. Basically a rootsofunity situation; then each $A_q$ can be a multiple of the proper coefficient of a cyclotomic polynomial. That does 'solve' the problem, in that you get the $A_q$'s in terms of the $a_q$, up to a single free variable. But of course you can also express a finite linear combination of many roots of unity in the same way, thus getting as many free variables as you wish. If the $a_q$ have some structure, then you might be able to say something  thought the problem often reduces to some nasty diophantine equations OR to requiring Schanuel's conjecture to separate things out. Your question is likely to get closed soon unless you make it a lot more precise and motivated. 

