Noncommutative 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 h
Pontryagin duality
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative
Elias M. Stein
Elias Menachem Stein (January 13, 1931 – December 23, 2018) was an American mathematician who was a leading figure in the field of harmonic analysis. He was the Albert Baldwin Dod Professor of Mathema
Gauss separation algorithm
Carl Friedrich Gauss, in his treatise Allgemeine Theorie des Erdmagnetismus, presented a method, the Gauss separation algorithm, of partitioning the magnetic field vector, B, measured over the surface
Riemann–Hilbert problem
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existe
Dyadic cubes
In mathematics, the dyadic cubes are a collection of cubes in Rn of different sizes or scales such that the set of cubes of each scale partition Rn and each cube in one scale may be written as a union
Kakeya set
In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane
Analyst's traveling salesman theorem
The analyst's traveling salesman problem is an analog of the traveling salesman problem in combinatorial optimization. In its simplest and original form, it asks which plane sets are subsets of rectif
Herz–Schur multiplier
In the mathematical field of representation theory, a Herz–Schur multiplier (named after Carl S. Herz and Issai Schur) is a special kind of mapping from a group to the field of complex numbers.
Tannaka–Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, betwee
Riesz transform
In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension d > 1. They are a type of singular integ
Trombi–Varadarajan theorem
In mathematics, the Trombi–Varadarajan theorem, introduced by Trombi and Varadarjan, gives an isomorphism between a certain space of spherical functions on a semisimple Lie group, and a certain space
Van der Corput lemma (harmonic analysis)
In mathematics, in the field of harmonic analysis,the van der Corput lemma is an estimate for oscillatory integralsnamed after the Dutch mathematician J. G. van der Corput. The following result is sta
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 theor
Cotlar–Stein lemma
In mathematics, in the field of functional analysis, the Cotlar–Stein almost orthogonality lemma is named after mathematicians Mischa Cotlarand Elias Stein. It may be used to obtain information on the
Constant-Q transform
In mathematics and signal processing, the constant-Q transform and variable-Q transform, simply known as CQT and VQT, transforms a data series to the frequency domain. It is related to the Fourier tra
Positive harmonic function
In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle. This result, the Herglotz-Ries
Oscillator representation
In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of th
Weakly symmetric space
In mathematics, a weakly symmetric space is a notion introduced by the Norwegian mathematician Atle Selberg in the 1950s as a generalisation of symmetric space, due to Élie Cartan. Geometrically the s
Weil–Brezin Map
In mathematics, the Weil–Brezin map, named after André Weil and Jonathan Brezin, is a unitary transformation that maps a Schwartz function on the real line to a smooth function on the Heisenberg manif
Muckenhoupt weights
In mathematics, the class of Muckenhoupt weights Ap consists of those weights ω for which the Hardy–Littlewood maximal operator is bounded on Lp(dω). Specifically, we consider functions f on Rn and
Minakshisundaram–Pleijel zeta function
The Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introduced by Subbaramiah Minakshisundaram and Åke Plei
List of harmonic analysis topics
This is a list of harmonic analysis topics. See also list of Fourier analysis topics and list of Fourier-related transforms, which are more directed towards the classical Fourier series and Fourier tr
Orlicz space
In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the Lp spaces. Like the Lp spaces, they are B
Harmonic (mathematics)
In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of
Bounded mean oscillation
In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of functions
Harish-Chandra's Ξ function
In mathematical harmonic analysis, Harish-Chandra's Ξ function is a special spherical function on a semisimple Lie group, studied by Harish-Chandra . Harish-Chandra used it to define Harish-Chandra's
Spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific
Set of uniqueness
In mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series. Their study is a relatively pure branch of harmonic analysis.
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, an
Applied and Computational Harmonic Analysis
Applied and Computational Harmonic Analysis is a bimonthly peer-reviewed scientific journal published by Elsevier. The journal covers studies on the applied and computational aspects of harmonic analy
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 mos
Wiener's Tauberian theorem
In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L
Singular integral operators of convolution type
In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions; equivalently they are the singular int
Gelfand pair
In mathematics, a Gelfand pair is a pair (G,K) consisting of a group G and a subgroup K (called an Euler subgroup of G) that satisfies a certain property on restricted representations. The theory of G
Locally compact quantum group
In mathematics and theoretical physics, a locally compact quantum group is a relatively new C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf
Arithmetic combinatorics
In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.
Fourier integral operator
In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operat
Orbital integral
In mathematics, an orbital integral is an integral transform that generalizes the spherical mean operator to homogeneous spaces. Instead of integrating over spheres, one integrates over generalized sp
Singular integral
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral o
Radial function
In mathematics, a radial function is a function defined on a Euclidean space Rn whose value at each point depends only on the distance between that point and the origin. For example, a radial function
Singular integral operators on closed curves
In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert
Sobolev spaces for planar domains
In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems
Pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differentia
Waldspurger formula
In representation theory of mathematics, the Waldspurger formula relates the special values of two L-functions of two related admissible irreducible representations. Let k be the base field, f be an a
Group algebra of a locally compact group
In functional analysis and related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra)
Harmonious set
In mathematics, a harmonious set is a subset of a locally compact abelian group on which every weak character may be uniformly approximated by strong characters. Equivalently, a suitably defined dual
Tide-Predicting Machine No. 2
Tide-Predicting Machine No. 2, also known as Old Brass Brains, was a special-purpose mechanical computer that uses gears, pulleys, chains, and other mechanical components to compute the height and tim
Oscillatory integral operator
In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form where the function S(x,y) is called the phase of the operator and the function a
Fourier algebra
Fourier and related algebras occur naturally in the harmonic analysis of locally compact groups. They play an important role in the duality theories of these groups. The Fourier–Stieltjes algebra and
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. This measure
Bateman transform
In the mathematical study of partial differential equations, the Bateman transform is a method for solving the Laplace equation in four dimensions and wave equation in three by using a line integral o
Harish-Chandra's Schwartz space
In mathematical abstract harmonic analysis, Harish-Chandra's Schwartz space is a space of functions on a semisimple Lie group whose derivatives are rapidly decreasing, studied by Harish-Chandra . It i
Zonal spherical function
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the mat
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 Lp space L2+ε(G) for any ε > 0.
Poisson boundary
In mathematics, the Poisson boundary is a measure space associated to a random walk. It is an object designed to encode the asymptotic behaviour of the random walk, i.e. how trajectories diverge when
Riemann–Lebesgue lemma
In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L1 function vanishes at infinity. It is of imp
Tube domain
In mathematics, a tube domain is a generalization of the notion of a vertical strip (or half-plane) in the complex plane to several complex variables. A strip can be thought of as the collection of co
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 Fourie
Hardy–Littlewood maximal function
In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis. It takes a locally integrable function f : Rd → C and returns