EQUATIONS
E-ISSN: 2944-9146 An Open Access International Journal of Mathematical and Computational Methods in Science and Engineering
Volume 3, 2023
The Plancherel Theory and the Uncertainty Principle
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Abstract: 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 $$ (1 + δ)^{p} μ_{G} (T) \hat{μ} (U) ≥ (1 − ε − δ)^{p}$$, where T is ε-concentration for $$ ψ ∈ L^{p} (G);$$ econd, we establish several variants of the statement that a function and its Fourier transform cannot be too rapidly decreasing namely $$ |ψ (g)| ≤ c_{1} exp ( −c_{2} ||g||^{2}) and ||π (ψ, u, \hat{u} )|| ≤ \tilde{c}_{1} (u) exp ( −\tilde{c}_{ 2} || \hat{u} ||^{2})$$ for all g ∈ G, on semisimple Lie group with the finite center.
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Keywords: Hausdorff groups, Heisenberg principle, uncertainty principle, Fourier transform, Wigner function, compact
group, Peter-Weyl theorem
Pages: 44-49
DOI: 10.37394/232021.2023.3.6