When transferring coefficients of a fourier series through an LTI system, each value of $a_k\$ is multiplied by $H(\jmath \omega)\$ in the system output. Therefore...

$\displaystyle a_k \rightarrow a_k \cdot H(\jmath \omega)$

The image below illustrates the process of taking the value of the frequency response function at each frequency of the coefficients and then multiplying by that value to yield the transformed coefficient values.

Note that when $H(\jmath \omega)$ is negative, the sign of the value of the coefficient is flipped.

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Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.