This is a list of
harmonic analysis topics. See also
list of Fourier analysis topics {{Short description, none
This is a list of Fourier analysis topics. See also the list of Fourier-related transforms, and the list of harmonic analysis topics.
Fourier analysis
* Multiplier (Fourier analysis)
* Fourier shell correlation
* Pinsk ...
and
list of Fourier-related transforms
This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized in t ...
, which are more directed towards the classical
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
and
Fourier transform of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
,
mathematical physics and
engineering.
Fourier analysis
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
*
Periodic function
*
Trigonometric function
*
Trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
**
Exponential sum
In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function
:e(x) = \exp(2\pi ix).\,
Therefore, a typic ...
*
Dirichlet kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as
D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac,
where is any nonn ...
*
Fejér kernel In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót ...
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Gibbs phenomenon
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Parseval's identity In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which ...
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Parseval's theorem
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Weyl differintegral
In mathematics, the Weyl integral (named after Hermann Weyl) is an operator defined, as an example of fractional calculus, on functions ''f'' on the unit circle having integral 0 and a Fourier series. In other words there is a Fourier series for ...
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Generalized Fourier series
**
Orthogonal functions In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the ...
**
Orthogonal polynomials
**
Empirical orthogonal functions
*
Set of uniqueness
Fourier transform
*
Continuous Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
*
Fourier inversion theorem
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Plancherel's theorem
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Convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
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Convolution theorem
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...
*
Positive-definite function
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Poisson summation formula
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Paley-Wiener theorem
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Sobolev space
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Time–frequency representation
*
Quantum Fourier transform
Topological groups
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Topological abelian group
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Haar measure
*
Discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comp ...
**
Dirichlet character
*
Amenable group
In mathematics, an amenable group is a locally compact topological group ''G'' carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely addit ...
**
Von Neumann's conjecture
*
Pontryagin duality
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Kronecker's theorem on diophantine approximation
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Almost periodic function
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Haral ...
*
Bohr compactification
*
Wiener's tauberian theorem
Representation theory
{{main, List of representation theory topics
*
Representation of a Lie group
*
Unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
*
Irreducible representation
*
Harish-Chandra character In mathematics, the Harish-Chandra character, named after Harish-Chandra, of a representation of a semisimple Lie group ''G'' on a Hilbert space ''H'' is a distribution on the group ''G'' that is analogous to the character of a finite-dimensional ...
*
Restricted representation In group theory, restriction forms a representation of a subgroup using a known representation of the whole group. Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to under ...
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Induced representation
*
Peter–Weyl theorem
In mathematics, 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. It was initially proved by Hermann Weyl, with his student Fritz Pete ...
*
Spherical harmonic
*
Casimir operator
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...
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Hecke operator
*
Stone–von Neumann theorem In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named afte ...
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Discrete series representation
*
Tempered representation
*
Langlands program
Fast Fourier transform
*
Bluestein's FFT algorithm
*
Cooley–Tukey FFT algorithm
The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size N = N_1N_2 in terms of ''N''1 ...
*
Rader's FFT algorithm
*
Number-theoretic transform
In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring.
Definition
Let R be any ring, let n\geq 1 be an inte ...
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Irrational base discrete weighted transform
In mathematics, the irrational base discrete weighted transform (IBDWT) is a variant of the fast Fourier transform using an irrational base; it was developed by Richard Crandall ( Reed College), Barry Fagin ( Dartmouth College) and Joshua Doeni ...
Analysis of unevenly spaced data
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Least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally ...
Applications
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FFT multiplication
*
Spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis function ...
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Fourier transform spectroscopy
Fourier-transform spectroscopy is a measurement technique whereby spectra are collected based on measurements of the coherence of a radiative source, using time-domain or space-domain measurements of the radiation, electromagnetic or not. It c ...
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Signal analysis
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Analytic signal
In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components. The real and imaginary parts of an analytic signal are real-valued functions related to each other by the Hilbe ...
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Welch method
Welch's method, named after Peter D. Welch, is an approach for spectral density estimation.
It is used in physics, engineering, and applied mathematics for estimating the power of a signal at different frequencies.
The method is based on the c ...
Harmonic analysis
Harmonic analysis
Harmonic analysis