Gelfand pairs over hypergroup joins

Vati, Kedumetse

doi:10.1007/s10474-019-00946-1

Cited by 6 publications

(5 citation statements)

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“…Applying the Naimark equivalent, there is a quotient representation π (v, ṽ) of π (v, ṽ) to close quotient subset V 1 /V 0 , there exists a densely define intertwining operator (π, V π ) → (π (v, ṽ) , V 1 /V 0 ) on the domain of which π (ψ ρ, ρ) = 0, from the properties of quotient representation, by continuity, we have π (ψ ρ, ρ) = 0 on V π . By summing over all ρ and ρ, we have π (ψ) = 0 for all representations of G, applying Plancherel theory, we obtain ψ = 0 [19].…”

Section: The Generalized Heisenberg Principlementioning

confidence: 99%

“…The equality (19) expresses the unitarity of the Fourier transform and can be rewritten in the form of the statement that for each F (ζ), the inverse Fourier transform is given by…”

Section: The Plancherel Theorymentioning

confidence: 99%

“…since if we take ζ = φ * * ψ and g = 1 then we obtain (19). Below we will follow notations of the David L. Donoho and Philip B. Stark [2] when it is convenient.…”

Section: The Plancherel Theorymentioning

confidence: 99%

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The Plancherel Theory and the Uncertainty Principle

Yaremenko

2023

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In this article, we are revising the Plancherel theory for a unimodular locally compact Hausdorff group with a Haar measure. Let be a connected semisimple real Lie group such that there exists an analytic diffeomorphism from the manifold to group according to the rule , decomposition is the Iwasawa decomposition of the group , dim(A) = rank(G). The group center(G) ⊂ K is closed, under the adjoint representation of is a maximal compact subgroup of the adjoint of ; subgroups and are simply connected. The associated minimal parabolic subgroup of is . Let and be Lies algebras of and , respectively, the norms correspond to the and dual algebra relative to the inner product induced by the Killing form of .Let an irreducible unitary representation of being presented as a left translation on where is finite-dimensional. Let be an element of the complexification of .Loosely said the Hardy uncertainty principle maintains that the function and its Fourier transform cannot be simultaneously both rapidly decreasing. The uncertainty principle is considered from several points of view: first, we consider the uncertainty principle in the case of a locally compact Hausdorff group G equipped with a probabilistic Haar measure μ G and K be a maximal compact subgroup of G with a probabilistic Haar measure μ K then we establish (1 + δ) p μ G (T) 𝜇̂ (U) ≥ (1 − ε − δ) p , where T is ε-concentration for ψ ∈ L p (G); second, we establish several variants of the statement that a function and its Fourier transform cannot be too rapidly decreasing namely |ψ (g)| ≤ c 1 exp ( −c 2 ∥g∥ 2 ) and ∥π (ψ, u, 𝑢 ̂)∥ ≤ 𝑐1 (u) exp ( −𝑐2 ∥ 𝑢 ̂∥2 ) for all g ∈ G, on semisimple Lie group with the finite center.

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Section: The Generalized Heisenberg Principlementioning

confidence: 99%

“…The equality (19) expresses the unitarity of the Fourier transform and can be rewritten in the form of the statement that for each F (ζ), the inverse Fourier transform is given by…”

Section: The Plancherel Theorymentioning

confidence: 99%

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The Plancherel Theory and the Uncertainty Principle

Yaremenko

2023

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“…One has the obvious consequence that the mapping f −→ f sets up a topological isomorphism between the topological vector spaces C (G) and C(G//K) (see [8,9]). So, for any…”

Section: Notations and Preliminariesmentioning

confidence: 99%

“…In fact, if K is a compact sub-hypergroup of a hypergroup G, the pair (G, K) is said to be a Gelfand pair if M c (G//K) the convolution algebra of measures with compact support on G//K is commutative. The notion of Gelfand pairs for hypergroups is well-known (see [3,8,9]). The goal of this paper is to extend Jewett work's by obtaining a Plancherel theorem over Gelfand pair associated with non-commutative hypergroup.…”

Section: Introductionmentioning

confidence: 99%