harmonic analysis

TheInfoList

Harmonic analysis is a branch of
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
concerned with the representation of
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
or signals as the superposition of basic
wave In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

s, and the study of and generalization of the notions of
Fourier series In mathematics, a Fourier series () is a periodic function composed of harmonically related Sine wave, sinusoids combined by a weighted summation. With appropriate weights, one cycle (or ''period'') of the summation can be made to approximate an ...
and
Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression o ...
s (i.e. an extended form of
Fourier analysis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
). In the past two centuries, it has become a vast subject with applications in areas as diverse as
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777– ...

,
representation theory Representation theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ( ...
,
signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniques c ...

,
quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quan ...
, tidal analysis and
neuroscience Neuroscience is the science, scientific study of the nervous system. It is a Multidisciplinary approach, multidisciplinary science that combines physiology, anatomy, molecular biology, developmental biology, cytology, computer science and Mathem ...

. The term "
harmonic of a vibrating string are harmonics. A harmonic is any member of the harmonic series (music), harmonic series. The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and ...
s" originated as 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 (), Dark Ages () ...
word ''harmonikos'', meaning "skilled in music". In physical
eigenvalue In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes, but the term has been generalized beyond its original meaning. 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
distributionDistribution may refer to: Mathematics *Distribution (mathematics) Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distr ...
of
compact support Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations, pacts are usually between two or more sovereign state A sovereign state is a po ...
(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 a very elementary form of an
uncertainty principle In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quant ...

in a harmonic-analysis setting. Fourier series can be conveniently studied in the context of
Hilbert space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s, which provides a connection between harmonic analysis and
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
. There are four versions of the Fourier Transform, dependent on the spaces that are mapped by the transformation (discrete/periodic-discrete/periodic: Digital Fourier Transform, continuous/periodic-discrete/aperiodic: Fourier Analysis, discrete/aperiodic-continuous/periodic: Fourier Synthesis, continuous/aperiodic-continuous/aperiodic: continuous Fourier Transform).

# Abstract harmonic analysis

One of the most modern branches of harmonic analysis, having its roots in the mid-20th century, is
analysis Analysis is the process of breaking a complexity, complex topic or Substance theory, 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 Ari ...
on
topological group 350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition In mathematics, a topological group is a group ...
s. The core motivating ideas are the various
Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression o ...
s, which can be generalized to a transform of
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
s defined on Hausdorff locally compact topological groups. The theory for abelian
locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact space, locally compact and Hausdorff space, Hausdorff. Locally compact groups are important because many examples of groups tha ...
s is called
Pontryagin duality 300px, The p-adic integer, 2-adic integers, with selected corresponding characters on Prüfer group, their Pontryagin dual group In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that a ...
. 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 In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
s. For general non-abelian 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 A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
Non-commutative harmonic analysis In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, ...
. 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#REDIRECT Plancherel theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematic ...
). However, many specific cases have been analyzed, for example SL''n''. In this case,
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It co ...
in infinite
dimensions thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...
play a crucial role.

# Other branches

*Study of the
eigenvalue In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

s and
eigenvector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

s of the
Laplacian In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
s,
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s, and (to a lesser extent)
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
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 In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression o ...
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, first defined by the mathematician Daniel Bernoulli Daniel Bernoulli Fellows of the Royal Society, FRS (; – 27 March 1782) was a Swiss people, Swiss mathematician and physicist and was one of the many prominent mathematicia ...
s and
spherical harmonic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s. * Harmonic analysis on tube domains is concerned with generalizing properties of
Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . I ...
s to higher dimensions.

# 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 (U.S.), low tide occurs roughly at moonrise and high tide with a high Moon, corresponding to the simple gravity model of two tidal bulges; at most places however, the Moon and tides have a phase shift. Tides are the rise and fall of sea level ...

s and vibrating
strings String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * Strings (1991 film), ''Strings'' (1991 fil ...
are common and simple examples. The theoretical approach is often to try to describe the system by a
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives ...

or
system of equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 major 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
Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression o ...
, the result of which is shown in the lower figure. Note that there is a prominent peak at 55 Hz, but that there are 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
harmonic of a vibrating string are harmonics. A harmonic is any member of the harmonic series (music), harmonic series. The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and ...
s.

*
Convergence of Fourier seriesIn mathematics, the question of whether the Fourier series of a periodic function convergent series, converges to the given function (mathematics), function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. ...
* Harmonic (mathematics) *
Spectral density estimation In statistical signal processing, the goal of spectral density estimation (SDE) is to estimate the spectral density (also known as the power spectral density) of a random signal from a sequence of time samples of the signal. Intuitively speaki ...
* Tate's thesis

# Bibliography

*Elias M. Stein, Elias Stein and Guido Weiss, ''Introduction to Fourier Analysis on Euclidean Spaces'', Princeton University Press, 1971. *Elias M. Stein, Elias Stein with Timothy S. Murphy, ''Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals'', Princeton University Press, 1993. *Elias M. Stein, 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 ℤ₂.) * 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.