The Fourier Series is used in both Mathematics and Electrical Engineering. The concept itself is Mathematical but the application of the Fourier Series strongly relates to Electrical Engineering in the area of Signal Analysis. This is an area of considerable overlap between Linear Algebra and Signal Analysis. One may study this from three perspectives:
General Fourier Series. (general orthogonal functions; inner products)
Trigonometric Fourier Series (sine and cosine series)
Exponential Fourier Series (complex exponentials ejw )