Abstract harmonic analysis
83 ideas in this subject. The map below lists the main ideas of the subject by area, in teaching order. 47 of its 152 ideas have a page so far.
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- ConceptLocally compact group
A locally compact group is a Hausdorff topological group whose topology is locally compact. These groups carry a Haar measure, an invariant way to measure and integrate, which makes harmonic analysis on them possible. They include all Lie groups, all discrete groups, all compact groups and the groups of points of algebraic groups over local fields such as .
- ConceptLocally compact abelian group
A locally compact abelian group, or LCA group, is a commutative topological group whose topology is Hausdorff and locally compact. These groups are the natural setting of Fourier analysis. Each LCA group has a Haar measure and a dual group of characters, and Pontryagin duality says that the group is recovered from its dual.
- ConceptHaar measure
A Haar measure on a locally compact group is a nonzero Radon measure that does not change under left translation by elements of the group. Every locally compact group has one, and it is unique up to a positive constant. It extends Lebesgue measure on and counting measure on discrete groups, and it is the basic tool for integration and harmonic analysis on groups.
- ConceptModular function
The modular function of a locally compact group measures how far a left Haar measure is from being right-invariant. It is a continuous homomorphism from the group to the multiplicative group of positive reals. It appears in every formula that moves a Haar integral across a right translation or an inversion. It is identically exactly for unimodular groups.
- ConceptUnimodular group
A unimodular group is a locally compact group whose left Haar measure is also right-invariant, that is, whose modular function is identically . Abelian groups, compact groups, discrete groups, semisimple Lie groups and connected nilpotent Lie groups are unimodular. Many results of harmonic analysis, such as the existence of lattices or the simplest form of trace formulas, require unimodularity.
- ConceptQuasi-invariant measure
A measure is quasi-invariant under a group action if moving by any group element keeps the same null sets. The measure itself may change, but only by a density. This weaker form of invariance is what one finds on homogeneous spaces without invariant measures and on infinite-dimensional spaces.
Map of the subject
Basic notions 2 of 4
- abstract harmonic analysis concept
- noncommutative harmonic analysis concept
- Locally compact group definition
- Locally compact abelian group definition
Haar measure and measures on groups 4 of 14
- Haar measure definition
- left-invariant measure definition
- existence of Haar measure statement
- uniqueness of Haar measure up to scaling statement
- Modular function definition
- Unimodular group definition
- normalized Haar measure on a compact group definition
- Haar measure on the circle example
- Lebesgue measure as Haar measure example
- Haar measure on the p-adic numbers example
- Quasi-invariant measure definition
- invariant measure on a homogeneous space statement
- Weil's quotient integral formula statement
- Lp spaces on groups definition
Group algebras 2 of 12
- convolution on a locally compact group definition
- group algebra L1(G) definition
- measure algebra M(G) definition
- bounded approximate identity of L1(G) statement
- Wendel's theorem on multipliers of L1(G) statement
- Group C*-algebra definition
- reduced group C*-algebra definition
- Group von Neumann algebra definition
- Fourier algebra definition
- Fourier–Stieltjes algebra definition
- Leptin's theorem statement
- Johnson's theorem on amenable group algebras statement
Characters and Pontryagin duality 2 of 11
- character of a locally compact abelian group definition
- dual group definition
- Pontryagin duality theorem statement
- dual of a compact group is discrete statement
- dual groups of the integers and the real line example
- annihilator duality statement
- structure theorem for locally compact abelian groups statement
- self-dual group example
- Gelfand spectrum of L1(G) statement
- character group definition
- Bohr compactification definition
Fourier analysis on locally compact abelian groups 4 of 11
- Fourier transform on locally compact abelian groups definition
- Fourier–Stieltjes transform of a measure definition
- Riemann–Lebesgue lemma for abelian groups statement
- Fourier inversion on abelian groups statement
- Plancherel theorem for abelian groups statement
- uniqueness theorem for Fourier–Stieltjes transforms statement
- Poisson summation formula for abelian groups statement
- Bochner's theorem statement
- Wiener's Tauberian theorem statement
- Herglotz's theorem on positive definite sequences statement
- Cohen's idempotent theorem statement
Finite abelian groups 2 of 9
- Fourier transform on finite groups definition
- additive character definition
- characters of a finite abelian group form a basis statement
- orthogonality of characters statement
- number of characters of a finite abelian group statement
- a finite abelian group is isomorphic to its dual statement
- Dirichlet character example
- Fourier analysis on the Boolean cube concept
- Gauss sum example
Positive definite functions 3 of 7
- Positive-definite function on a group definition
- Gelfand–Raikov theorem statement
- positive-definite kernel definition
- conditionally negative definite function definition
- Schoenberg's theorem on negative definite kernels statement
- Haagerup property definition
- Kazhdan's property (T) definition
Spectral synthesis and thin sets 0 of 4
- spectral synthesis concept
- Malliavin's theorem on failure of spectral synthesis statement
- Rajchman measure definition
- Helson–Lowdenslager theorem statement
Almost periodic functions on groups 0 of 4
- almost periodic function on a group definition
- invariant mean on almost periodic functions statement
- von Neumann's theory of almost periodic functions on groups statement
- weakly almost periodic function definition
Amenability and invariant means 4 of 19
- Amenable group definition
- Invariant mean definition
- Banach limit example
- Følner sequence definition
- Følner condition implies amenability statement
- Reiter's condition statement
- Hulanicki's theorem statement
- Kesten's criterion for amenability statement
- Markov–Kakutani fixed-point theorem statement
- Day's fixed point theorem statement
- amenability of abelian and solvable groups statement
- elementary amenable group definition
- non-amenability of free groups statement
- paradoxical decomposition definition
- Tarski's theorem on paradoxical groups statement
- Banach–Tarski paradox statement
- von Neumann conjecture conjecture
- Ol'shanskii's counterexample to the von Neumann conjecture counterexample
- amenable semigroup definition
Representations in harmonic analysis 10 of 24
- Unitary representation definition
- left regular representation definition
- Induced representation definition
- system of imprimitivity definition
- Mackey imprimitivity theorem statement
- Mackey machine concept
- Fell topology definition
- type I group definition
- Fourier transform on a nonabelian group definition
- Plancherel measure definition
- Plancherel theorem for unimodular type I groups statement
- Heisenberg group example
- Stone–von Neumann theorem statement
- Weil representation definition
- orbit method concept
- Kirillov character formula statement
- tempered representation definition
- Discrete series representation definition
- Principal series representation definition
- Harish-Chandra character definition
- Harish-Chandra's regularity theorem statement
- Harish-Chandra's Schwartz space definition
- Harish-Chandra Plancherel theorem statement
- representation theory of SL2(R) concept
Harmonic analysis on compact groups 6 of 12
- Peter–Weyl theorem statement
- matrix coefficient definition
- Schur orthogonality relations statement
- orthonormality of irreducible characters statement
- Complete reducibility of representations of compact groups statement
- Fourier series on a compact group concept
- Plancherel theorem for compact groups statement
- Weyl integration formula statement
- Tannaka–Krein duality statement
- harmonic analysis on SU(2) example
- spherical harmonics as representations of SO(3) example
- Wigner D-matrix definition
Homogeneous spaces, Gelfand pairs and spherical functions 5 of 16
- Homogeneous space definition
- harmonic analysis on homogeneous spaces concept
- Gelfand pair definition
- Gelfand's trick statement
- Hecke algebra of a locally compact group definition
- zonal spherical function definition
- Harish-Chandra's formula for spherical functions statement
- spherical transform definition
- Plancherel theorem for spherical functions statement
- Harish-Chandra c-function definition
- Gindikin–Karpelevich formula statement
- Helgason Fourier transform definition
- Paley–Wiener theorem for symmetric spaces statement
- Harish-Chandra isomorphism statement
- Zonal spherical harmonics definition
- Addition theorem for spherical harmonics statement
Automorphic and arithmetic harmonic analysis 3 of 5
- Selberg trace formula statement
- Arthur–Selberg trace formula statement
- Automorphic form definition
- Tate's thesis statement
- harmonic analysis on the adeles concept
More ideas in this subject
Concepts 22
- Approximate identity
- Besicovitch almost periodic function
- Bohr almost periodic function
- Følner condition
- Amenable Banach algebra
- Discrete Fourier transform
- Exponential functions on semigroups
- Harmonic analysis
- Almost periodic function
- Amenable group action
- Convolution
- Fourier–Stieltjes transform
- Functional equations on groups
- Unitary dual
- Wiener algebra
- Harmonic analysis on groups
- Fourier monomial on the circle
- Group algebra of a locally compact group
- Positive-definite function
- Spherical function
- Character of an abelian group
- Pontryagin dual
Theorems and results 6
Counterexamples 1
Problems 1
Definitions 33
- Approximate identity
- Besicovitch almost periodic function
- Bohr almost periodic function
- Følner condition
- Amenable Banach algebra
- Amenable group
- Discrete Fourier transform
- Exponential functions on semigroups
- Harmonic analysis
- Zonal spherical harmonics
- Almost periodic function
- Amenable group action
- Convolution
- Fourier–Stieltjes transform
- Functional equations on groups
- Haar measure
- Locally compact abelian group
- Positive-definite function on a group
- Unitary dual
- Bohr compactification
- Fourier transform on locally compact abelian groups
- Invariant mean
- Wiener algebra
- Gelfand pair
- Harmonic analysis on groups
- Unitary representation
- Fourier monomial on the circle
- Group algebra of a locally compact group
- Positive-definite function
- Spherical function
- Character of an abelian group
- Pontryagin dual
- Modular function