Fourier Series vs Taylor Series – Are They Inverses?

fourier seriessequences-and-seriestaylor expansion

I've understood Taylor series as being the representation of a "transcendental" function, using power functions with coefficents represented by appropriate derivatives. (Or maybe it is the MacLauren series, where $\cos x= 1-\frac{x^2}{2!}+\frac{x^4}{4!}+…$)

I've understood Fourier series as being the representation of periodic linear functions using integral coefficients multiplied by transcendental functions.

Is my understanding correct in either or both cases? And if one consists of derivatives and the other of integrals, does that mean that Fourier series are the converse (or "inverse") of Taylor series?

Best Answer

Just a brief comparison:

Fourier series are:

  • Global in nature. Fourier series are computed using an integral over one period (they represent the entire function over one period even if it is discontinuous, piece-wise continuous, etc.... Gibbs phenomenon not withstanding)
  • Fourier series decompose a function by representing it as a linear combination of basis functions (sine and cosine). These basis functions are orthogonal.
  • Fourier series are invertable. That is once you have your Fourier coefficients you can reconstruct the entire function from the coefficients (up to a point, i.e. Gibbs phenomenon).

On the other hand:

Taylor series are:

  • Local in nature. Taylor series are computed using an infinite number of derivatives at one point (therefore they cannot represent functions which are discontinuous, piece-wise continuous, etc).
  • Taylor series decompose a function by representing it as a fixed combination of derivatives. These "basis" functions are not orthogonal.
  • Taylor series are invertable only in the neighborhood of a point. You cannot, in general, recover the entire function from a Taylor series.
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