exponential sum - general cases [closed]

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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closed as not a real question by Felipe Voloch, fedja, Andrés E. Caicedo, Gerry Myerson, Ryan BudneyDec 5 '11 at 4:23

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Is the sum finite or infinite? Is it just one equation or a system? – fedja Dec 4 '11 at 12:47
finite. one equation. so we have one equation with $n$ unknowns. that is, $a_q$ (or $A_q$) for $q \in \{1,...,n\}$ – Sean Dec 4 '11 at 14:21
You should indicate if the $A_q$ and $a_q$ are assumed real (which they likely are). What about $k$ and $s$ ? – Jacques Carette Dec 4 '11 at 14:25
in the general case $k$ is complex, $s$ is real. – Sean Dec 4 '11 at 14:39
Is there something I'm missing? How can you hope to solve one equation for N variables? – Antoine Levitt Dec 4 '11 at 14:57

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 roots-of-unity 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.