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I found the coefficients of the Fourier series. The partial sum of the Fourier series is defined on the interval $[0,2\pi]$ by

$$S_N(x) = \frac{a_0}{2} + \sum_{n=1}^{N}[ a_n \cos(nx) + b_n \sin(nx)],$$

where

$$a_0 =\frac{1}{\pi} \int_{0}^{2\pi} f(x)\, dx, $$ $$a_n = \frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos(nx)\, dx,$$ $$b_n =\frac{1}{\pi} \int_{0}^{2\pi} f(x) \sin(nx)\, dx.$$

Is there any way to put all the coefficients in the partial sum and to derive the function? I struggled after I put everything in one equation to see if it can be simplified more.

Your help is greatly appreciated!

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  • $\begingroup$ I am new to this website, I try to fix the way to write equations. I am so sorry $\endgroup$
    – Riddle2795
    Dec 29, 2021 at 9:04
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    $\begingroup$ Have a look at this. $\endgroup$
    – Gary
    Dec 29, 2021 at 9:10
  • $\begingroup$ Thank you ! @Gary $\endgroup$
    – Riddle2795
    Dec 29, 2021 at 9:56
  • $\begingroup$ At the moment it is not clear what you are asking for. What would you like to achive/do exactly? $\endgroup$
    – Gary
    Dec 29, 2021 at 10:05
  • $\begingroup$ if I plug a0,an,and bn into S_N(x). How could be simplified more ? Can we solve the sum or S_n(x) be written in simple way ? $\endgroup$
    – Riddle2795
    Dec 29, 2021 at 15:42

1 Answer 1

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I found the above question might be a classic issue. Its digital form is called compressed-sensing.
Here's a link and page 6 introduces this problem.

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