# Harmonic analysis

Harmonic analysis

Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called "harmonics"(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"n" is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions. 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 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 spaces, 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 analysis on topological groups. The core motivating idea are the various Fourier transforms, which can be generalized to a transform of functions defined on Hausdorff locally compact topological groups.

The theory for abelian locally compact groups is called Pontryagin duality; it is considered to be in a satisfactory stateFact|date=September 2008, as far as explaining the main features of harmonic analysis goes.

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 groups.

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 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 SL"n". In this case, it turns out that representations in infinite dimension play a crucial role.

Other branches

*Study of the eigenvalues and eigenvectors of the Laplacian on domains, manifolds, and (to a lesser extent) graphs 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"n" 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 functions and spherical harmonics. See the book reference.
* Harmonic analysis on tube domains is concerned with generalizing properties of Hardy spaces to higher dimensions.

ee 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

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### См. также в других словарях:

• harmonic analysis — n. 1. the study of Fourier series 2. the act of breaking a periodic function into components, each expressed as a sine or cosine function …   English World dictionary

• harmonic analysis — Math. 1. the calculation of Fourier series and their generalization. 2. the study of Fourier series and their generalization. Also called Fourier analysis. [1865 70] * * * ▪ mathematics       mathematical procedure for describing and analyzing… …   Universalium

• harmonic analysis — noun analysis of a periodic function into a sum of simple sinusoidal components • Syn: ↑Fourier analysis • Hypernyms: ↑analysis * * * noun : the approximate expression of a periodic function known only for some values of the independent variable… …   Useful english dictionary

• harmonic analysis — harmoninė analizė statusas T sritis fizika atitikmenys: angl. harmonic analysis; harmonical analysis vok. harmonische Analyse, f rus. гармонический анализ, m pranc. analyse harmonique, f …   Fizikos terminų žodynas

• harmonic analysis — noun A study of the representation of functions or signals as the superposition of basic waves, involving the notions of harmonic functions, trigonometric series, Fourier series, Fourier transforms, almost periodic functions, and others …   Wiktionary

• harmonic analysis — noun Date: 1867 the expression of a periodic function as a sum of sines and cosines and specifically by a Fourier series …   New Collegiate Dictionary

• harmonic analysis — harmon′ic anal′ysis n. math. the calculation or study of Fourier series and their generalization • Etymology: 1865–70 …   From formal English to slang

• Spherical harmonic analysis — Harmonic Har*mon ic (h[aum]r*m[o^]n [i^]k), Harmonical Har*mon ic*al ( [i^]*kal), a. [L. harmonicus, Gr. armoniko s; cf. F. harmonique. See {Harmony}.] 1. Concordant; musical; consonant; as, harmonic sounds. [1913 Webster] Harmonic twang! of… …   The Collaborative International Dictionary of English

• Noncommutative harmonic analysis — In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups which are not commutative. Since for locally compact abelian groups have a well understood theory, Pontryagin… …   Wikipedia

• List of harmonic analysis topics — This is a list of harmonic analysis topics, by Wikipedia page. See also list of Fourier analysis topics and list of Fourier related transforms, which are more directed towards the classical Fourier series and Fourier transform of mathematical… …   Wikipedia

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