Fourier transform

Summary

The Fourier transform writes a function on $\R^n$ as a superposition of plane waves $e^{2\pi i x \cdot \xi}$ and records the amplitude of each frequency $\xi$. It turns derivatives into multiplication by polynomials in $\xi$, and convolution into ordinary multiplication. So it changes linear partial differential equations with constant coefficients into algebraic or ordinary differential equations. It extends to $L^2$ functions and to tempered distributions. See the definition.