Harmonic analysis
14147
221027129
2008-06-22T18:54:06Z
143.205.128.119
'''Harmonic analysis''' is the branch of [[mathematics]] that studies the representation of functions or signals as the superposition of basic [[wave]]s. It investigates and generalizes the notions of [[Fourier series]] and [[Fourier transform]]s. The basic waves are called "[[harmonic]]s"(in physics), hence the name "harmonic analysis," but the name "harmonic" in this context is generalized beyond its original meaning of integer frequency multiples. In the past two centuries, it has become a vast subject with applications in areas as diverse as [[signal processing]], [[quantum mechanics]], and [[neuroscience]].
The classical Fourier transform on '''R'''<sup>''n''</sup> is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as [[tempered distribution]]s. For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The [[Paley-Wiener theorem]] is an example of this. 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. This is a very elementary form of an [[uncertainty principle]] in a harmonic analysis setting. See also [[classic harmonic analysis]].
Fourier series can be conveniently studied in the context of [[Hilbert space]]s, which provides a connection between harmonic analysis and [[functional analysis]].
==Abstract harmonic analysis==
One of the more modern branches of harmonic analysis, having its roots in the mid-twentieth century, is [[mathematical analysis|analysis]] on [[topological group]]s. The core motivating idea are the various [[Fourier transform]]s, which can be generalized to a transform of [[function (mathematics)|function]]s defined on [[local compactness|locally compact Hausdorff groups]].
The theory for [[abelian group|abelian]] locally [[compact group]]s is called [[Pontryagin duality]]; it is considered to be in a satisfactory state, as far as explaining the main features of harmonic analysis goes. It is developed in detail on its dedicated page.
Harmonic analysis studies the properties of that duality and Fourier transform; and attempts to extend those features to different settings, for instance to the case of non-abelian [[Lie group]]s.
For general nonabelian locally compact groups, harmonic analysis is closely related to the theory of unitary group representations. 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. This choice of harmonics enjoys some of the useful 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.
If the group is neither abelian nor compact, no general satisfactory theory is currently known. By "satisfactory" one would mean ''at least'' the equivalent of [[Plancherel theorem]]. However, many specific cases have been analyzed, for example [[General linear group|SL<sub>''n''</sub>]]. In this case, it turns out that [[Group representation|representations]] in infinite dimension play a crucial role.
==Other branches==
*Study of the [[eigenvalue]]s and [[eigenvector]]s of the [[Laplacian]] on [[domain (mathematics)|domain]]s, [[manifold]]s, and (to a lesser extent) [[Graph (mathematics)|graph]]s is also considered a branch of harmonic analysis. See e.g., [[hearing the shape of a drum]].
* 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 invariant to rotations. Decomposing the Fourier transform to its radial and spherical components leads to topics such as [[Bessel function]]s and [[spherical harmonic]]s. See the book reference.
* Harmonic analysis on tube domains is concerned with generalizing properties of [[Hardy space]]s to higher dimensions.
==See also==
* [[Fourier series]]
==References==
[[Elias M. Stein]] and Guido Weiss, ''Introduction to Fourier Analysis on Euclidean Spaces'', Princeton University Press, 1971. ISBN 0-691-08078-X
[[Yitzhak Katznelson]], ''An introduction to harmonic analysis'', Third edition. Cambridge University Press, 2004. ISBN 0-521-83829-0; 0-521-54359-2
[[Category:Mathematical analysis]]
[[Category:Harmonic analysis|*]]
[[bg:Хармоничен анализ]]
[[de:Fourieranalyse]]
[[es:Análisis armónico]]
[[eo:Fourier-a analizo]]
[[fr:Analyse harmonique (mathématiques)]]
[[it:Analisi armonica]]
[[mt:Analisi Armonika]]
[[nl:Fourieranalyse]]
[[pl:Analiza harmoniczna]]
[[pt:Análise harmónica]]
[[ru:Гармонический анализ]]
[[tr:Fourier analizi]]
[[zh:傅立叶分析]]