In dynamical systems with linear differential equations, we almost always break up the function of independent variable in sines and cosines.
But suppose that my function is smooth and periodic. Then what advantage do I get by using Fourier series instead Taylor?
Inside the radius of convergence, the Taylor series converges uniformly which is favourable, while the Fourier might only converge pointwise.
So why use Fourier? I’d find using polynomials as more intuitive than sines or cosines.
Best Answer
The complex exponentials are eigenfunctions of the derivative and integral operators. So if you're analyzing linear differential equations, and using Fourier series, then you can consider each term on its own. If you use Taylor series you have to consider interactions between one term and other terms in the series. (This is also why we often write our Fourier series in terms of complex exponentials rather than sines and cosines)
Extrapolation. If I have a function $f(x)$ and I approximate it on a region $[x_1, x_2]$ with a finite-length Taylor series $F_T(x)$, then outside of $[x_1, x_2]$ the Taylor series will tend to go to infinity. If I approximate it with a finite-length Fourier series, the series will remain bounded as $x\to\infty$.