Editing Harmonic analysis

{{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.

== Development of harmonic analysis ==
Historically, [[harmonic function]]s first referred to the solutions of [[Laplace's equation]].<ref><!-- DO NOT USE CITATION TEMPLATES FOR THIS REFERENCE! They are unable to handle the Fall 2020–2021 date specified in the reference. -->Burtscher, Annegret (Fall 2020–2021). "[https://www.math.ru.nl/~burtscher/lecturenotes/2021PDEnotes.pdf Introduction to Partial Differential Equations, Course module NWI-WB046B]" (PDF). Radboud University Nijmegen. Retrieved 2025-01-19.</ref> This terminology was extended to other [[special functions]] that solved related equations,<ref>{{cite book|title=Special functions and the theory of group representation|author=N. Vilenkin|year=1968}}</ref> then to [[Eigenfunction|eigenfunctions]] of general [[Elliptic operator|elliptic operators]],<ref>{{See also|Atiyah-Singer index theorem}}</ref> and nowadays harmonic functions are considered as a generalization of periodic functions<ref>{{cite web | url=https://www.britannica.com/science/harmonic-analysis | title=Harmonic analysis &#124; Mathematics, Fourier Series & Waveforms &#124; Britannica }}</ref> in [[function space]]s defined on [[manifold|manifolds]], for example as solutions of general, not necessarily [[elliptic partial differential equation|elliptic]], [[partial differential equations]] including some [[boundary conditions]] that may imply their symmetry or periodicity.<ref name=tao>{{citation|url=https://www.math.ucla.edu/~tao/247a.1.06f/notes0.pdf|first=Terence|last=Tao|author-link=Terence Tao|title=Harmonic Analysis|work=MATH 247A : Fourier analysis|publisher=University of California, Los Angeles|access-date=2025-01-19}}</ref>

== Fourier analysis ==

{{main|Fourier analysis}}
The classical [[Fourier transform]] on '''[[Real number|R]]'''<sup>''n''</sup> is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as [[distribution (mathematics)#Tempered distributions and Fourier transform|tempered distributions]]. For instance, if we impose some requirements on a distribution ''f'', we can attempt to translate these requirements into the Fourier transform of ''f''. The [[Paley–Wiener theorem]] is an example. The Paley–Wiener theorem immediately implies that if ''f'' is a nonzero [[Distribution (mathematics)|distribution]] of [[compact 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 an [[uncertainty principle]] in a harmonic-analysis setting.

Fourier series can be conveniently studied in the context of [[Hilbert space]]s, which provides a connection between harmonic analysis and [[functional analysis]]. There are four versions of the Fourier transform, dependent on the spaces that are mapped by the transformation:
* Discrete/periodic–discrete/periodic: [[Discrete Fourier transform]]
* Continuous/periodic–discrete/aperiodic: [[Fourier series]]
* Discrete/aperiodic–continuous/periodic: [[Discrete-time Fourier transform]]
* Continuous/aperiodic–continuous/aperiodic: [[Fourier transform]]
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==
Abstract harmonic analysis is primarily concerned with how real or
complex-valued [[function (mathematics)|function]]s (often on very general domains) can be studied using symmetries such
as [[translations]] or [[rotations]] (for instance via the [[Fourier transform]] and its relatives); this field is of
course related to real-variable harmonic analysis, but is perhaps closer in spirit to [[representation theory]] and [[functional analysis]].<ref name=tao/>

One of the most modern branches of harmonic analysis, having its roots in the mid-20th century, is [[mathematical analysis|analysis]] on [[topological group]]s. The core motivating ideas are the various [[Fourier transform]]s, which can be generalized to a transform of functions defined on Hausdorff [[locally compact group|locally compact topological groups]].<ref>{{cite book|title=Introduction to the Representation Theory of Compact and Locally Compact Groups|author=Alain Robert}}</ref>

One of the major results in the theory of functions on [[abelian group|abelian]] locally compact groups is called [[Pontryagin 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-abelian [[Lie group]]s.<ref>{{cite book|title=A Course in Abstract Harmonic Analysis|author=Gerald B Folland}}</ref>

Harmonic analysis is closely related to the theory of unitary group representations for general non-abelian locally compact groups. For compact groups, the [[Peter–Weyl theorem]] explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations.<ref>{{cite book|title=Introduction to the Representation Theory of Compact and Locally Compact Groups|author=Alain Robert}}</ref> 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 underlying [[Group (mathematics)|group]] 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 the [[Plancherel theorem]]). However, many specific cases have been analyzed, for example, [[Special linear group|SL<sub>''n''</sub>]]. In this case, [[Group representation|representations]] in infinite [[Dimension (mathematics and physics)|dimensions]] play a crucial role.

==Applied harmonic analysis==

[[File:Bass Guitar Time Signal of open string A note (55 Hz).png|thumb|400 px| Bass-guitar time signal of open-string A note (55&nbsp;Hz)]]
[[File:Fourier Transform of bass guitar time signal.png|thumb|400 px| Fourier transform of bass-guitar time signal of open-string A note (55&nbsp;Hz)<ref>{{Cite web |date=2015-07-07 |title=A More Accurate Fourier Transform |url=https://sourceforge.net/projects/amoreaccuratefouriertransform/ |access-date=2024-08-26 |website=SourceForge |language=en}}</ref>]]

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.  Ocean [[tide]]s and vibrating [[String (music)|strings]] are common and simple examples.  The theoretical approach often tries to describe the system by a [[differential equation]] or [[system 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 to [[Data collection|acquire 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&nbsp;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 the [[Fourier transform]], shown in the lower figure.  There is a prominent peak at 55&nbsp;Hz, but other peaks at 110&nbsp;Hz, 165&nbsp;Hz, and at other frequencies corresponding to integer multiples of 55&nbsp;Hz.  In this case, 55&nbsp;Hz is identified as the fundamental frequency of the string vibration, and the integer multiples are known as [[harmonic]]s.

==Other branches==
*Study of the [[eigenvalue]]s and [[eigenvector]]s of the [[Laplacian]] on [[domain (mathematical analysis)|domain]]s, [[manifold]]s, and (to a lesser extent) [[Graph (discrete mathematics)|graph]]s is also considered a branch of harmonic analysis. See, e.g., [[hearing the shape of a drum]].<ref>{{cite book |last1=Terras |first1=Audrey |title=Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane |date=2013 |publisher=Springer |location=New York, NY |isbn=978-1461479710 |page=37 |edition=2nd |url=https://books.google.com/books?id=LcHBAAAAQBAJ&q=harmonic+analysis+hear+shape+of+a+drum&pg=PA37 |access-date=12 December 2017}}</ref>
* Harmonic analysis on Euclidean spaces deals with properties of the [[Fourier transform]] on '''R'''<sup>''n''</sup> that have no analog on general groups. For example, the fact that the Fourier transform is rotation-invariant. Decomposing the Fourier transform into its radial and spherical components leads to topics such as [[Bessel function]]s and [[spherical harmonic]]s.
* Harmonic analysis on tube domains is concerned with generalizing properties of [[Hardy space]]s to higher dimensions.
* [[Automorphic forms]] are generalized harmonic functions, with respect to a symmetry group. They are an old and at the same time active area of development in harmonic analysis due to their connections to the [[Langlands program]].
* Non linear harmonic analysis is the use of harmonic and [[functional analysis]] tools and techniques to study [[nonlinear systems]]. This includes both problems with infinite [[degrees of freedom]] and also non linear [[Operator (mathematics)|operators]] and [[partial differential equations|equations]].<ref>{{cite book | chapter-url=https://www.degruyter.com/document/doi/10.1515/9781400882090-002/html?lang=en | doi=10.1515/9781400882090-002 | chapter=Non-Linear Harmonic Analysis, Operator Theory and P.d.e. | title=Beijing Lectures in Harmonic Analysis. (AM-112) | date=1987 | last1=Coifman | first1=R. R. | last2=Meyer | first2=Yves | pages=1–46 | isbn=978-1-4008-8209-0 }}</ref>

== Major results ==
{{empty section|date=May 2024}}

==See also==
* [[Convergence of Fourier series]]
* [[Fourier analysis]] for computing periodicity in evenly-spaced data
* [[Harmonic (mathematics)]]
* [[Least-squares spectral analysis]] for computing periodicity in unevenly spaced data
* [[Spectral density estimation]]
* [[Tate's thesis]]

==References==
{{Reflist}}

==Bibliography==
*[[Elias M. Stein|Elias Stein]] and [[Guido Weiss]], ''Introduction to Fourier Analysis on Euclidean Spaces'', [[Princeton University Press]], 1971. {{ISBN|0-691-08078-X}}
*[[Elias M. Stein|Elias Stein]] with Timothy S. Murphy, ''Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals'', Princeton University Press, 1993.
*[[Elias M. Stein|Elias Stein]], ''Topics in Harmonic Analysis Related to the Littlewood-Paley Theory'', Princeton University Press, 1970.
*[[Yitzhak Katznelson]], ''An introduction to harmonic analysis'', Third edition. Cambridge University Press, 2004. {{ISBN|0-521-83829-0}}; 0-521-54359-2
* [[Terence Tao]], [https://www.math.ucla.edu/~tao/preprints/fourier.pdf Fourier Transform]. (Introduces the decomposition of functions into odd + even parts as a harmonic decomposition over <math>\mathbb{Z}_2</math>.)
* Yurii I. Lyubich. ''Introduction to the Theory of Banach Representations of Groups''. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988.
* [[George W. Mackey]], [https://doi.org/10.1090/S0273-0979-1980-14783-7 Harmonic analysis as the exploitation of symmetry–a historical survey], ''Bull. Amer. Math. Soc.'' 3 (1980), 543–698.
* M. Bujosa, A. Bujosa and A. Garcıa-Ferrer. [http://dx.doi.org/10.1109/TSP.2015.2469640 Mathematical Framework for Pseudo-Spectra of Linear Stochastic Difference Equations], ''IEEE Transactions on Signal Processing'' vol. 63 (2015), 6498–6509.

==External links==
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