Harmonic Analysis

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

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

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