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Fourier series approximate given functions using sums of "sinx" curves with differing frequency.

Taylor series approximate given functions using sums of power functions with differing degree.

Obviously, both are used in their own individual way, but which series is normally more efficient to use? For example, when approximating a "y=sinx" function, obviously the Fourier series will be considerably easier. However, when approximating a polynomial curve, such as "y=x^2", the Taylor Series "approximation" is much more simple. (I believe the F. S. for y=x^2 is given in this document: http://quantum.phys.unm.edu/E02087.pdf)

However, functions obviously encompass far, far more than just power functions or trigonometric functions. Hence, the question is, for a function which is neither purely trigonometric or purely "power series", which Series (of approximation) is more efficient in general? Sorry for the poor formatting and my horrible understanding of mathematics (if it does exist)

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  • $\begingroup$ For a function to have Taylor series that converges to the function itself, the function has to be analytic (en.wikipedia.org/wiki/Analytic_function). However, much more general functions admit Fourier series. $\endgroup$ – Taisuke Yasuda Aug 16 '18 at 1:24
  • $\begingroup$ If the function isn't periodic and you wish to approximate it on the real line, Fourier series won't even work, whereas Taylor still might. $\endgroup$ – dbx Aug 16 '18 at 3:04

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