Fourier series vs polynomials in approximation

Question

I am looking into uses of Fourier series. I learned that it can be used to approximate functions (link).

However, I think there are ways to approximate functions with polynomials (not limited to Taylor series).

What advantage does the Fourier series have over polynomials in approximation?

I think there is a disadvantage to the Fourier series. While the way to compute the values of polynomials is obvious, computing the values of Fourier series needs approximate values of the trigonometric functions.

Answer 1

One advantage with Fourier series is that due to the frequency domain specification, you can have series approximation to different levels of precision by including lesser or more terms. For polynomial approximation you have to recompute the coefficients. In Fourier series approximations you can just truncate the higher frequency terms.

Also, many operations are more efficient in frequency domain. E.g., convolution, multiplication etc., so much so that even polynomial multiplication may be mapped to use Fast Fourier Transforms for efficiency.

Answer 2

You’re absolutley right, you can approximate functions with polynomials, Fourier series, and other methods! As far as why you would pick a fourier series over a taylor series is heavily dependent on the function you are approximating. The main advantage to the fourier series is its periodicity. It helps to visualize how each behaves:

-Taylor: https://images.app.goo.gl/QD6h7eoRZcn1U2Th7

-Fourier: https://images.app.goo.gl/sovp1UKxtMjAidgh8

In the first case, the Taylor expansion provides accurate approximations of functions over a range limited by the order(N) of the series.

In the second case, the Fourier series provides an infinite range to the function, though the acurracy of the method is limited by the order (N) of the series.