Notes on Fourier series

The trigonometric Fourier series is a beautiful mathematical theory that shows how to decompose a periodic function into an infinite sum of sinusoids. These are my notes on the subject, with some examples and the connection to linear algebra in Hilbert space. Coefficients of Fourier series Let’s assume that is a well-behaved -periodic [1] function and that we can find coefficients and such that: Then we say that the Fourier series on the right-hand side converges to . We’ll talk more about the assumptions mentioned above and convergence in the next section. Note that when , the sum becomes just ; therefore it’s customary to write the series starting with , with a separate constant component (which is the function's average over one period). To make computations nicer, this constant is typically called , so: Our goal is to find the coefficients and that satisfy this equation. We’ll do this in three steps. Step 1: Integrate both sides of the equation between and [2]. Per Appendix A, all…