Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms. In the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, and neuroscience.
The term “harmonics” originated in physical eigenvalue problems, to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics on stringed musical instruments, but the term has been generalized beyond its original meaning.
The classical Fourier transform on Rn is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions. For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley–Wiener theorem is an example of this. The Paley–Wiener theorem immediately implies that if f is a nonzero distribution of compact support (these include functions of compact support), then its Fourier transform is never compactly supported. This is a very elementary form of an uncertainty principle in a harmonic analysis setting. See also Convergence of Fourier series.
Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis.