Harmonic analysis is a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

concerned with investigating the connections between a function and its representation in

frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...

. The frequency representation is found by using the

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

for functions on unbounded domains such as the full

real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...

or by

Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...

for functions on bounded domains, especially

periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...

s on finite 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 Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

, tidal analysis, spectral analysis, and

neuroscience Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions, and its disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, ...

. The term "

harmonic In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...

s" originated from the

Ancient Greek Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek ...

word ''harmonikos'', meaning "skilled in music". In physical

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes. Still, the term has been generalized beyond its original meaning.

Development of harmonic analysis

Historically,

harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that i ...

s first referred to the solutions of Laplace's equation. This terminology was extended to other special functions that solved related equations, then to

eigenfunctions In mathematics, an eigenfunction of a linear map, linear operator ''D'' defined on some function space is any non-zero function (mathematics), function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor calle ...

of general elliptic operators, and nowadays harmonic functions are considered as a generalization of periodic functions in

function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...

s defined on manifolds, for example as solutions of general, not necessarily elliptic,

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...

including some boundary conditions that may imply their symmetry or periodicity.

Fourier analysis

The classical

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 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 of

compact support In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed ...

(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 The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...

in a harmonic-analysis setting. Fourier series can be conveniently studied in the context of

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

s, which provides a connection between harmonic analysis and

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

. 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 In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...

* Continuous/periodic–discrete/aperiodic:

* Discrete/aperiodic–continuous/periodic: Discrete-time Fourier transform * Continuous/aperiodic–continuous/aperiodic:

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 functions (often on very general domains) can be studied using symmetries such as translations or rotations (for instance via the

and its relatives); this field is of course related to real-variable harmonic analysis, but is perhaps closer in spirit to

and

. One of the most modern branches of harmonic analysis, having its roots in the mid-20th century, is

analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...

s. The core motivating ideas are the various

s, which can be generalized to a transform of functions defined on Hausdorff locally compact topological groups. One of the major results in the theory of functions on 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 In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

s. 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. 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 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, SL_''n''. In this case, representations in infinite dimensions play a crucial role.

Applied harmonic analysis

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 Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another. Tide tables ...

s and vibrating 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 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 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

, shown in the lower figure. There is a prominent peak at 55 Hz, but other peaks at 110 Hz, 165 Hz, and at other frequencies corresponding to integer multiples of 55 Hz. In this case, 55 Hz is identified as the fundamental frequency of the string vibration, and the integer multiples are known as

Other branches

*Study of the

s and

eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...

s of the

Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...

on domains,

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

s, 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

on R^''n'' 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 Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...

s and spherical harmonics. * Harmonic analysis on tube domains is concerned with generalizing properties of Hardy spaces 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 In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by . It seeks to relate the structure of Galois groups in algebraic number t ...

. * Non linear harmonic analysis is the use of harmonic and

tools and techniques to study nonlinear systems. This includes both problems with infinite

degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...

and also non linear operators and equations.

Major results

References

Bibliography

* Elias Stein and Guido Weiss, ''Introduction to Fourier Analysis on Euclidean Spaces'',

Princeton University Press Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial ...

, 1971. * Elias Stein with Timothy S. Murphy, ''Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals'', Princeton University Press, 1993. * 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. ; 0-521-54359-2 * Terence Tao
Fourier Transform
(Introduces the decomposition of functions into odd + even parts as a harmonic decomposition over

\mathbb_2

.) * 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
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
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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