Harmonic analysis

Summary

Harmonic analysis studies functions by writing them as superpositions of basic waves, and studies operators through their action on these waves. Its classical core is the theory of Fourier series on the circle and of the Fourier transform on $\R^n$. On a general locally compact group the waves are replaced by characters or irreducible representations, which gives harmonic analysis on groups. The subject supplies tools for partial differential equations, number theory, probability and signal processing.