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I trying to solve for a function inside a finite summation and haven't been able to find many resources on the topic. Below is the general form of the equation and I'd like to rearrange it so solve for the function $f(n)$. It seems like there should be some theorem about this form but I can't seem to find anything. Thanks.

$$x=\sum_{n=1}^{N}g(n)f(n)$$

brad14
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  • Is the equation you give for a fixed $N$, or is it true for any $N$? – Paul Apr 01 '20 at 13:32
  • For the expression above, N would be a predetermined fixed value. It would not be true for any random N. – brad14 Apr 01 '20 at 13:55
  • Is there anything specific you can say about $g(n)$? – Paul Apr 01 '20 at 13:56
  • $g(n)$ is a list of coefficients which gives a different coefficient for each $n$ and $f(n)$ is an analytic function that is also a function of $n$. This is essentially the summation of coefficients multiplied by an analytic function. – brad14 Apr 01 '20 at 14:00
  • For a fixed $N$ and $g(n)$ there are infinite possibilities for $f(n)$. It is underdetermined. – Paul Apr 01 '20 at 14:14

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