In a previous note , we showed how to linearize trigonometric polynomials using complex numbers, Euler's formula, and DeMoivre's formula. Here we explain why this gives, in fact, the Fourier series of these trigonometric polynomials.
"An Algebraic Remark on the Fourier Series of a Trigonometric Polynomial." Missouri J. Math. Sci. 15 (2) 94 - 96, Spring 2003. https://doi.org/10.35834/2003/1502094