Harmonic analysis

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Harmonic analysis

Harmonic analysis is the branch of mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
that studies the representation of functions or signals as the superposition of basic wave

Wave

A wave is a disturbance that propagates through space and time, usually with transference of energy. While a mechanical wave exists in a medium , waves of electromagnetic radiation can travel through vacuum, that is, without a medium....
s. It investigates and generalizes the notions of Fourier series

Fourier series

In mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions, namely sine wave . The study of Fourier series is a branch of Fourier analysis....
and Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions....
s. The basic waves are called "harmonic

Harmonic

In acoustics and telecommunication, a harmonic of a wave is a component frequency of the Signalling that is an integer multiple of the fundamental frequency....
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

Signal processing

Signal processing is the analysis, interpretation, and manipulation of signal . Signals of interest include: audio signal processing, , time-varying measurement values and sensor data, for example biological data such as electrocardiograms, control system signals, telecommunication transmission signals such as radio signals, and many others....
, quantum mechanics

Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
, and neuroscience

Neuroscience

Neuroscience is a field devoted to the scientific study of the nervous system. The Society for Neuroscience was founded in 1969, but the study of the brain started a long time ago....
. The classical Fourier transform on Rⁿ is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distribution

Tempered distribution

*Distribution *Tempered representation ...
s.

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Encyclopedia

Harmonic analysis is the branch of mathematics

Mathematics

Wave

Fourier series

Fourier transform

Harmonic

Signal processing

Quantum mechanics

Neuroscience

Tempered distribution

*Distribution *Tempered representation ...
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

Distribution (mathematics)

In mathematical analysis, distributions are objects which generalize function s. They extend the concept of derivative to all locally integrable functions and beyond, and are used to formulate generalized solutions of partial differential equations....
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

Uncertainty principle

In quantum physics, the Werner Heisenberg uncertainty principle states that certain physical quantities, like the position and momentum, cannot both have precise values at the same time....
in a harmonic analysis setting. See also classic harmonic analysis.

Fourier series can be conveniently studied in the context of Hilbert space

Hilbert space

The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
s, which provides a connection between harmonic analysis and functional analysis

Functional analysis

Functional analysis is the branch of mathematics, and specifically of mathematical analysis, concerned with the study of vector spaces and operators acting upon them....
.

Abstract harmonic analysis

One of the more modern branches of harmonic analysis, having its roots in the mid-twentieth century, is analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit , whether the limit of a sequence or the limit of a function....
on topological group

Topological group

In mathematics, a topological group is a group G together with a topological space on G such that the group's binary operation and the group's inverse function are continuous function ....
s. The core motivating idea are the various Fourier transform

Fourier transform

Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
s defined on Hausdorff locally compact topological groups.

The theory for abelian

Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
locally compact group

Compact group

In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion....
s is called Pontryagin duality

Pontryagin duality

In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform....
; it is considered to be in a satisfactory state, 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 group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
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

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
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

Plancherel theorem

In mathematics, the Plancherel theorem is a result in harmonic analysis, first proved by Michel Plancherel [1]. In its simplest form it states that if a function is in both Lp space and Lp space, then its Fourier transform is in L2; moreover the Fourier transform map is isometric....
. However, many specific cases have been analyzed, for example SL_n

Special linear group

In mathematics, the special linear group of degree n over a field F is the set of n×n Matrix with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
. In this case, it turns out that representations

Group representation

In the mathematics field of representation theory, group representations describe abstract group in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrix so that the group operation can be represented by matrix multiplication....
in infinite dimension play a crucial role.

Other branches

Study of the eigenvalues and eigenvectors of the Laplacian on domain
Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
s, manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s, and (to a lesser extent) graph
Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
s is also considered a branch of harmonic analysis. See e.g., hearing the shape of a drum
Hearing the shape of a drum

To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of basic harmonic series , via the use of mathematics theory....
.
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 function
Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical#Mathematics solutions y of Bessel's differential equation:...
s and spherical harmonic
Spherical Harmonic

Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire series of books by Catherine Asaro which tells the story of Pharaoh Dyhianna Selei , ruler of the Skolian Empire, after the Radiance War fought by the Imperialate and their enemy Eubians....
s. See the book reference.
Harmonic analysis on tube domains is concerned with generalizing properties of Hardy space
Hardy space

In complex analysis, the Hardy spaces Hp are certain spaces of Holomorphic function on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G....
s to higher dimensions.