Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Harmonic analysis
open-in-new
<p>Harmonic analysis is a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a> concerned with investigating the connections between a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> and its representation in <a href="/facts/Frequency/QUSbPaW8">frequency</a>. The frequency representation is found by using the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> for functions on unbounded domains such as the full <a href="/facts/Real_line/6eq42xX5">real line</a> or by <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> for functions on bounded domains, especially <a href="/facts/Periodic_function/Sm8lJQsF">periodic functions</a> on finite <a href="/facts/Interval_(mathematics)/e9se3vTe">intervals</a>. Generalizing these transforms to other domains is generally called <a href="/facts/Fourier_analysis/sUA8JP6q">Fourier analysis</a>, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a>, <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, <a href="/facts/Tidal_analysis/C1RnsktH">tidal analysis</a>, <a href="/facts/Spectral_theory/8sAEEIMM">spectral analysis</a>, and <a href="/facts/Neuroscience/o5RV8Cas">neuroscience</a>. </p><p>The term "<a href="/facts/Harmonic/9U9REzeR">harmonics</a>" originated from the <a href="/facts/Ancient_Greek/LIK7NlUh">Ancient Greek</a> word <i>harmonikos</i>, meaning "skilled in music". In physical <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> problems, it began to mean waves whose frequencies are <a href="/facts/Multiple_(mathematics)/tkte6NXa">integer multiples</a> of one another, as are the frequencies of the <a href="/facts/Harmonic_series_(music)/PdElIrxL">harmonics of music notes</a>. Still, the term has been generalized beyond its original meaning. </p>