Editing Discrete Fourier transform (section)

===Partial differential equations===
Discrete Fourier transforms are often used to solve [[partial differential equations]], where again the DFT is used as an approximation for the [[Fourier series]] (which is recovered in the limit of infinite ''N''). The advantage of this approach is that it expands the signal in complex exponentials <math>e^{inx}</math>, which are eigenfunctions of differentiation: <math>{\text{d} \big( e^{inx} \big) }/\text{d}x = in e^{inx}</math>. Thus, in the Fourier representation, differentiation is simple—we just multiply by <math>in</math>.  (However, the choice of <math>n</math> is not unique due to aliasing; for the method to be convergent, a choice similar to that in the [[#Trigonometric interpolation polynomial|trigonometric interpolation]] section above should be used.) A [[linear differential equation]] with constant coefficients is transformed into an easily solvable algebraic equation. One then uses the inverse DFT to transform the result back into the ordinary spatial representation. Such an approach is called a [[spectral method]].