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", 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 Rⁿ 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A wave is a disturbance that propagates through space or spacetime, transferring energy and momentum and sometimes angular momentum. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... This article is about the components of sound. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Drawing of the cells in the chicken cerebellum by S. Ramón y Cajal Neuroscience is a field that is devoted to the scientific study of the nervous system. ... This page deals with mathematical distributions. ... In mathematics the Paley-Wiener theorem relates growth properties of entire functions on Cn and Fourier transformation of Schwartz distributions of compact support. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ... In mathematics, the question whether the Fourier series of a periodic function converges to the given function and in what sense is a rich field of research, sometimes called 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. The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

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 locally compact Hausdorff groups. Analysis has its beginnings in the rigorous formulation of calculus. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ...

The theory for abelian locally compact groups 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. In mathematics, an abelian group, also called a commutative group, is a group (G, * ) such that a * b = b * a for all a and b in G. In other words, the order in which the binary operation is performed doesnt matter. ... In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ...

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. In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...

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. The Peter-Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. ... This picture illustrates how the hours on a clock form a group under modular addition. ...

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. In mathematics, the Plancherel theorem is a result in harmonic analysis, first proved by Michel Plancherel. ... In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

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 Rⁿ 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.

In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... In mathematics, the domain of a function is the set of all input values to the function. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In mathematics, hearing the shape of a drum relates to a series of results that do just that, i. ... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: x2 for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred... Spherical Harmonic is a fantasy novel by Catherine Asaro which tells the story of Pharaoh Dyhianna (Dehya) Selei, ruler of the Skolian Imperialate, after the Radiance War fought by the Imperialate and their enemy Eubian Concord. ... In complex analysis, the Hardy spaces are analogues of the Lp spaces of functional analysis. ...

See also

• Fourier series

The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

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 Yithzak Katznelson (also transcribed Jizchak Katzenelson) was a Jewish teacher, poet and dramatist. ...