Editing Harmonic analysis (section)

{{Short description|Study of superpositions in mathematics}}
{{for|the process of determining the structure of a piece of music|Harmony}}
{{Use American English|date = March 2019}}
{{broader|Harmonic (mathematics)}}
'''Harmonic analysis''' is a branch of [[mathematics]] concerned with investigating the connections between a [[Function (mathematics)|function]] and its representation in [[frequency]]. The frequency representation is found by using the [[Fourier transform]] for functions on unbounded domains such as the full [[real line]] or by [[Fourier series]] for functions on bounded domains, especially [[periodic function]]s on finite [[Interval (mathematics)|intervals]]. Generalizing these transforms to other domains is generally called [[Fourier analysis]], although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as [[number theory]], [[representation theory]], [[signal processing]], [[quantum mechanics]], [[tidal analysis]], [[spectral theory|spectral analysis]], and [[neuroscience]].

The term "[[harmonic]]s" originated from the [[Ancient Greek]] word ''harmonikos'', meaning "skilled in music".<ref>[http://www.etymonline.com/index.php?term=harmonic "harmonic"]. ''[[Online Etymology Dictionary]]''.</ref> In physical [[eigenvalue]] problems, it began to mean waves whose frequencies are [[Multiple (mathematics)|integer multiples]] of one another, as are the frequencies of the [[Harmonic series (music)|harmonics of music notes]]. Still, the term has been generalized beyond its original meaning.