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# List of harmonic analysis topics

This is a list of harmonic analysis topics. [1]

73 relations: Almost periodic function, Amenable group, Analytic signal, Bohr compactification, Casimir element, Chirp Z-transform, Convolution, Convolution theorem, Cooley–Tukey FFT algorithm, Dirichlet character, Dirichlet kernel, Discrete Fourier transform, Discrete Fourier transform (general), Discrete series representation, Empirical orthogonal functions, Engineering, Exponential sum, Fast Fourier transform, Fejér kernel, Fourier analysis, Fourier inversion theorem, Fourier series, Fourier transform, Fourier-transform spectroscopy, Generalized Fourier series, Gibbs phenomenon, Haar measure, Harish-Chandra character, Harmonic analysis, Hecke operator, Induced representation, Irrational base discrete weighted transform, Irreducible representation, Kronecker's theorem, Langlands program, List of Fourier analysis topics, List of Fourier-related transforms, Mathematical analysis, Mathematical physics, Multiplication algorithm, Orthogonal functions, Orthogonal polynomials, Paley–Wiener theorem, Parseval's identity, Parseval's theorem, Periodic function, Peter–Weyl theorem, Plancherel theorem, Poisson summation formula, Pontryagin duality, ... Expand index (23 more) »

## 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".

## 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.

## Analytic signal

In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components.

## Bohr compactification

In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H. The concept is named after Harald Bohr who pioneered the study of almost periodic functions, on the real line.

## Casimir element

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.

## Chirp Z-transform

The Chirp Z-transform (CZT) is a generalization of the discrete Fourier transform.

## Convolution

In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.

## Convolution theorem

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the pointwise product of Fourier transforms.

## 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.

## Dirichlet character

In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z. Dirichlet characters are used to define Dirichlet ''L''-functions, which are meromorphic functions with a variety of interesting analytic properties.

## Dirichlet kernel

In mathematical analysis, the Dirichlet kernel is the collection of functions e^.

## 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 complex-valued function of frequency.

## Discrete Fourier transform (general)

In mathematics, the discrete Fourier transform over an arbitrary ring generalizes the discrete Fourier transform of a function whose values are complex numbers.

## Discrete series representation

In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²(G).

## Empirical orthogonal functions

In statistics and signal processing, the method of empirical orthogonal function (EOF) analysis is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data.

## Engineering

Engineering is the creative application of science, mathematical methods, and empirical evidence to the innovation, design, construction, operation and maintenance of structures, machines, materials, devices, systems, processes, and organizations.

## 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 Therefore, a typical exponential sum may take the form summed over a finite sequence of real numbers xn.

## Fast Fourier transform

A fast Fourier transform (FFT) is an algorithm that samples a signal over a period of time (or space) and divides it into its frequency components.

## Fejér kernel

In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series.

## Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

## Fourier inversion theorem

In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform.

## Fourier series

In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.

## Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

## 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 electromagnetic radiation or other type of radiation.

## Generalized Fourier series

In mathematical analysis, many generalizations of Fourier series have proved to be useful.

## Gibbs phenomenon

In mathematics, the Gibbs phenomenon, discovered by Available on-line at: and rediscovered by, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity.

## Haar measure

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

## 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 representation of a compact group.

## Harmonic analysis

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).

## Hecke operator

In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by, is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations.

## Induced representation

In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup to a representation of the (whole) group itself.

## 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 Doenias (NeXT Software) in the early 1990s using Mathematica.

## Irreducible representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper subrepresentation (\rho|_W,W), W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hermitian vector space V is the direct sum of irreducible representations.

## Kronecker's theorem

In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by.

## Langlands program

In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry.

## List of Fourier analysis topics

This is a list of Fourier analysis topics.

## List of Fourier-related transforms

This is a list of linear transformations of functions related to Fourier analysis.

## Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

## Mathematical physics

Mathematical physics refers to the development of mathematical methods for application to problems in physics.

## Multiplication algorithm

A multiplication algorithm is an algorithm (or method) to multiply two numbers.

## Orthogonal functions

In mathematics, orthogonal functions belong to a function space which is a vector space that has a bilinear form.

## Orthogonal polynomials

In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.

## Paley–Wiener theorem

In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform.

## 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.

## Parseval's theorem

In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.

## Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.

## 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.

## Plancherel theorem

In mathematics, the Plancherel theorem is a result in harmonic analysis, proven by Michel Plancherel in 1910.

## Poisson summation formula

In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform.

## Pontryagin duality

In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as \R, the circle, or finite cyclic groups.

## Positive-definite function

In mathematics, a positive-definite function is, depending on the context, either of two types of function.

## Quantum Fourier transform

In quantum computing, the quantum Fourier transform (for short: QFT) is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fourier transform.

Rader's algorithm (1968), named for Charles M. Rader of MIT Lincoln Laboratory, is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of prime sizes by re-expressing the DFT as a cyclic convolution (the other algorithm for FFTs of prime sizes, Bluestein's algorithm, also works by rewriting the DFT as a convolution).

## Representation of a Lie group

In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry.

## Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

## Restricted representation

In mathematics, restriction is a fundamental construction in representation theory of groups.

## Set of uniqueness

In mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series.

## Signal processing

Signal processing concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements.

## Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function itself and its derivatives up to a given order.

## Spectral method

Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, potentially involving the use of the Fast Fourier Transform.

## Spherical harmonics

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere.

## Stone–von Neumann theorem

In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators.

## Tempered representation

In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the L''p'' space for any ε &gt; 0.

## Time–frequency representation

A time–frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency.

## Topological abelian group

In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group.

## Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.

## Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

## 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.

## 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 is a locally compact (Hausdorff) topological group and the representations are strongly continuous.

## Von Neumann conjecture

In mathematics, the von Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators.

## Welch's method

In physics, engineering, and applied mathematics, Welch's method, named after P.D. Welch, is used for estimating the power of a signal at different frequencies: that is, it is an approach to spectral density estimation.

## Weyl integral

In mathematics, the Weyl integral is an operator defined, as an example of fractional calculus, on functions f on the unit circle having integral 0 and a Fourier series.

## Wiener's tauberian theorem

In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932.

## References

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