Harmonic analysis is a branch ofmathematics concerned with investigating the connections between afunction and its representation infrequency. The frequency representation is found by using theFourier transform for functions on unbounded domains such as the fullreal line or byFourier series for functions on bounded domains, especiallyperiodic functions on finiteintervals. Generalizing these transforms to other domains is generally calledFourier analysis, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse asnumber theory,representation theory,signal processing,quantum mechanics,tidal analysis,spectral analysis, andneuroscience.
The term "harmonics" originated from theAncient Greek wordharmonikos, meaning "skilled in music".[1] In physicaleigenvalue problems, it began to mean waves whose frequencies areinteger multiples of one another, as are the frequencies of theharmonics of music notes. Still, the term has been generalized beyond its original meaning.
Historically,harmonic functions first referred to the solutions ofLaplace's equation.[2] This terminology was extended to otherspecial functions that solved related equations,[3] then toeigenfunctions of generalelliptic operators,[4] and nowadays harmonic functions are considered as a generalization of periodic functions[5] infunction spaces defined onmanifolds, for example as solutions of general, not necessarilyelliptic,partial differential equations including someboundary conditions that may imply their symmetry or periodicity.[6]
The classicalFourier transform onRn is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such astempered distributions. For instance, if we impose some requirements on a distributionf, we can attempt to translate these requirements into the Fourier transform off. ThePaley–Wiener theorem is an example. The Paley–Wiener theorem immediately implies that iff is a nonzerodistribution ofcompact support (these include functions of compact support), then its Fourier transform is never compactly supported (i.e., if a signal is limited in one domain, it is unlimited in the other). This is an elementary form of anuncertainty principle in a harmonic-analysis setting.
Fourier series can be conveniently studied in the context ofHilbert spaces, which provides a connection between harmonic analysis andfunctional analysis. There are four versions of the Fourier transform, dependent on the spaces that are mapped by the transformation:
As the spaces mapped by the Fourier transform are, in particular, subspaces of the space of tempered distributions it can be shown that the four versions of the Fourier transform are particular cases of the Fourier transform on tempered distributions.
Abstract harmonic analysis is primarily concerned with how real orcomplex-valuedfunctions (often on very general domains) can be studied using symmetries suchastranslations orrotations (for instance via theFourier transform and its relatives); this field is ofcourse related to real-variable harmonic analysis, but is perhaps closer in spirit torepresentation theory andfunctional analysis.[6]
One of the most modern branches of harmonic analysis, having its roots in the mid-20th century, isanalysis ontopological groups. The core motivating ideas are the variousFourier transforms, which can be generalized to a transform of functions defined on Hausdorfflocally compact topological groups.[7]
One of the major results in the theory of functions onabelian locally compact groups is calledPontryagin duality. Harmonic analysis studies the properties of that duality. Different generalization of Fourier transforms attempts to extend those features to different settings, for instance, first to the case of general abelian topological groups and second to the case of non-abelianLie groups.[8]
Harmonic analysis is closely related to the theory of unitary group representations for general non-abelian locally compact groups. For compact groups, thePeter–Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations.[9] This choice of harmonics enjoys some of the valuable properties of the classical Fourier transform in terms of carrying convolutions to pointwise products or otherwise showing a certain understanding of the underlyinggroup structure. See also:Non-commutative harmonic analysis.
If the group is neither abelian nor compact, no general satisfactory theory is currently known ("satisfactory" means at least as strong as thePlancherel theorem). However, many specific cases have been analyzed, for example,SLn. In this case,representations in infinitedimensions play a crucial role.
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Many applications of harmonic analysis in science and engineering begin with the idea or hypothesis that a phenomenon or signal is composed of a sum of individual oscillatory components. Oceantides and vibratingstrings are common and simple examples. The theoretical approach often tries to describe the system by adifferential equation orsystem of equations to predict the essential features, including the amplitude, frequency, and phases of the oscillatory components. The specific equations depend on the field, but theories generally try to select equations that represent significant principles that are applicable.
The experimental approach is usually toacquire data that accurately quantifies the phenomenon. For example, in a study of tides, the experimentalist would acquire samples of water depth as a function of time at closely enough spaced intervals to see each oscillation and over a long enough duration that multiple oscillatory periods are likely included. In a study on vibrating strings, it is common for the experimentalist to acquire a sound waveform sampled at a rate at least twice that of the highest frequency expected and for a duration many times the period of the lowest frequency expected.
For example, the top signal at the right is a sound waveform of a bass guitar playing an open string corresponding to an A note with a fundamental frequency of 55 Hz. The waveform appears oscillatory, but it is more complex than a simple sine wave, indicating the presence of additional waves. The different wave components contributing to the sound can be revealed by applying a mathematical analysis technique known as theFourier transform, shown in the lower figure. There is a prominent peak at 55 Hz, but other peaks at 110 Hz, 165 Hz, and at other frequencies corresponding to integer multiples of 55 Hz. In this case, 55 Hz is identified as the fundamental frequency of the string vibration, and the integer multiples are known asharmonics.
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