Harmonic analysis is a branch of mathematics related to the representation of functions or signals as the overlaying of basic waves and the study and generalization of notions of Fourier series and Fourier transforms (ie an extended form of Fourier analysis ). In the last two centuries, it has become a great subject with applications in areas as diverse as numerical theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience.
The term "harmonics" originated as the ancient Greek word, "harmonikos", meaning "expert in music". In the problems of physical own values began to mean waves whose frequencies are integer multiples of each other, as well as the frequencies of the harmonics of musical notes, but the term has become generalized beyond its original meaning.
The classical Fourier transform in Rn is still an area of ongoing research, particularly in relation to the Fourier transform in more general objects such as tempered distributions. For example, if we impose some requirements on a distribution f, we can try to translate these requirements in terms of the Fourier transform of f. Paley-Wiener's theorem is an example of this. The Paley-Wiener theorem immediately implies that if f is a non-zero distribution of compact support (these include compact support functions), then its Fourier transform is never compactly supported. This is a very elementary form of a principle of uncertainty in a harmonic analysis setting. See also: Convergence of Fourier series. The Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis.