Section
Harmonic Analysis
Haar measure, convolution, Fourier analysis, positive-definite functions, and harmonic analysis on locally compact groups.
Core idea
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Harmonic analysis
Unitary-representation and character foundations
- Conjugate and contragredient unitary representations
- Continuous unitary character
- Hilbert direct sum of unitary representations
- Tensor product of unitary representations
Haar measure, convolution, and regular representations
- Approximate identity in L1(G)
- Convolution on a locally compact group
- Haar integral
- Haar measure
- Hecke algebra of a locally compact group pair
- Integrated operator of a continuous representation
- Integrated form of a unitary representation
- Involution on a group convolution algebra
- L1 group algebra
- Left and right regular representations of a locally compact group
- Measure algebra of a locally compact group
- Modular function of a locally compact group
- Quasi-regular representation
- Unimodular locally compact group
- Weil integration formula
- Young's convolution inequality
Fourier transform and Plancherel theory
- Dual Haar measure
- Fourier convolution theorem on a locally compact abelian group
- Fourier inversion theorem for locally compact abelian groups
- Fourier transform on a locally compact abelian group
- Hausdorff–Young inequality on a locally compact abelian group
- Plancherel measure for a type I locally compact group
- Plancherel theorem for locally compact abelian groups
- Pontryagin dual
- Pontryagin duality theorem
Fractal uncertainty and Beurling–Malliavin theory
- Bounded Fourier support
- Paley–Wiener theorem for bounded Fourier support
- Hilbert transform
- Poisson kernel of the upper half-plane
- Poisson extension to the upper half-plane
- Dirichlet-to-Neumann operator on the upper half-plane
- X-ray transform
- Oscillatory integral
- Beurling–Malliavin multiplier theorem
- Radial-line growth functional of a weight
- Linewise Poisson extension operator
- Exact plurisubharmonic Beurling–Malliavin extension
- Plurisubharmonic Beurling–Malliavin proposition
- Analytic Beurling–Malliavin proposition
- Higher-dimensional Beurling–Malliavin multiplier theorem
- Damping function for a frequency set
- Damping-function fractal uncertainty theorem
- Damping functions from line porosity
- Fractal uncertainty principle
- Higher-dimensional line-porous fractal uncertainty principle
See the dependency-ordered batch index for the associated porosity, several-complex-variables, and quantum-chaos knowls.
Positive-definite functions and duality
- Bochner's theorem for locally compact abelian groups
- Fell topology on the unitary dual
- Fourier algebra of a locally compact group
- Fourier–Stieltjes algebra
- GNS construction for a positive-definite function
- Kernel description of Fell topology
- Matrix coefficient of a unitary representation
- Positive-definite function on a group
- Unitary dual of a locally compact group
Amenability
- Amenability via weak containment of the trivial representation
- Amenable locally compact group
- Følner condition
- Invariant mean
- Reiter condition
- Weak containment of unitary representations
Induction and imprimitivity
- Locally compact homogeneous space
- Induced unitary representation
- Induction in stages for unitary representations
- Mackey imprimitivity theorem
- Mackey machine for semidirect products
- Quasi-invariant measure under a locally compact group action
- System of imprimitivity
- Unitary Frobenius reciprocity
Smooth representations of nonarchimedean groups
- Smooth representation of a totally disconnected group
- Admissible representation of a p-adic group
- Supercuspidal representation
- Unramified representation
- Normalized parabolic induction for a p-adic group
- Whittaker model
These provide the representation-theoretic input to the modern Langlands foundations.
Direct integrals and nonabelian Plancherel theory
- Direct integral of unitary representations
- Discrete and continuous spectrum of a unitary representation
- Multiplicity function of a representation
- Nonabelian group Fourier transform
- Plancherel theorem for unimodular type I groups
- Tempered dual of a locally compact group
- Tempered unitary representation
Bruhat and Harish-Chandra analysis
- Harish-Chandra Schwartz algebra
- Harish-Chandra Schwartz space
- Harish-Chandra Xi-function
- Restricted tensor product of local test-function spaces
- Schwartz–Bruhat space on a locally compact abelian group
- Schwartz–Bruhat space on the adeles
- Schwartz–Bruhat space over a local field
- Spherical Fourier transform
- Spherical function on a Gelfand pair
Periodic Fourier analysis and oscillatory waves
- Carrier frequency parameter
- Fourier character on the standard torus
- Periodic Fourier coefficient
- Fourier harmonic
- Orthogonality of periodic Fourier characters
- Mean of a product of periodic Fourier sums
- Periodic Fourier series
- Oscillatory modulation
- Oscillatory phase
- Oscillatory pulse
- Periodic antiderivative
- Periodic Fourier multiplier
- Pulse envelope
- Decay of smooth periodic Fourier coefficients
- Smooth periodic Fourier reconstruction
- Transverse polarization of a vector wave
- Wave amplitude
- Wavelength
- Wavenumber
- Local wavevector
- Zero Fourier mode
- Gaussian bound from a decreasing net growth rate
Torus covers and support combinatorics
- Diophantine direction
- Directional inverse with a Diophantine bound
- Normalized torus average
- Resonant mode of a directional derivative
- Small divisor
- Fourier criterion for descent under a torus covering
- Torus coverings preserve normalized averages
Sobolev spaces and pressure analysis
- Calderón–Zygmund kernel
- Lp boundedness of Calderón–Zygmund operators
- Calderón–Zygmund operator
- Kernel of a double Riesz transform in three dimensions
- Euclidean L2 Fourier multiplier
- Kernel cancellation in a multiplication commutator
- Principal-value singular integral
- Riesz transform
- Scaled cutoff commutator estimate