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Given a Müntz–Szàsz sequence of positive real numbers (λ k) k and a bounded interval I ⊂ R , Müntz–Szàsz theorem for completeness of the monomials { x λ k } k in L 2 (I) can be extended to a class of compact extensions of Heisenberg groups. The idea is to define infinitely many coordinate functions of generic unitary representations which are shown to be compactly supported and whose Fourier transforms extend analytically on the complex plane with suitable exponential domination. The representation theory and a Plancherel formula of reducible generic representations play an important role in the proofs. [ABSTRACT FROM AUTHOR]

Let G be a separable unimodular locally compact group of type I, and let N be a unimodular closed normal subgroup of G of type I, such that G/N is compact. Let for 1 < p ≤ 2 , ‖ F p (G) ‖ and ‖ F p (N) ‖ denote the norms of the corresponding L p -Fourier transforms. We show that ‖ F p (G) ‖ ≤ ‖ F p (N) ‖ . In the particular case where G = K ⋉ N is defined by a semi-direct product of a separable unimodular locally compact group N of type I and a compact subgroup K of the automorphism group of N, we show that equality holds if N has a K-invariant sequence (φ j) j of functions in L 1 (N) ∩ L p (N) such that ‖ F φ j ‖ q / ‖ φ j ‖ p tends to ‖ F p (N) ‖ when j goes to infinity. We show further that in some cases, an extremal function of N extends to an extremal function of G. [ABSTRACT FROM AUTHOR]

Given a strictly increasing sequence of positive integers ( n), the Müntz-Szàsz theorem for completeness of the monomials { x} in L([0, 1]) can be extended to Euclidean spin groups which are the universal coverings of Euclidean motion groups SO( n)⋉R. Towards such an objective, we rephrase the completeness condition in terms of an integral against the monomials { x} of the coordinates functions associated to the Garding vectors of G, whose Fourier transform are shown to admit an analytic continuation to the whole complex plane with an exponential domination. This allows us to formulate and prove several variants of Müntz-Szàsz's theorem in these settings. These upshots are proved using the Plancherel theory related to the group Fourier transform. [ABSTRACT FROM AUTHOR]

Let $$K$$ be a compact subgroup of automorphisms of $$\mathbb R ^n$$. We prove in this paper a generalization of Hardy's uncertainty principle on the semi-direct product $$K\ltimes \mathbb R ^n$$, and we solve the sharpness problem. As a consequence, a complete analogue of classical Hardy's theorem is obtained. The representation theory and the Plancherel formula play an important role in the proofs. [ABSTRACT FROM AUTHOR]

Let K be a compact subgroup of automorphisms of ℝ. We formulate and prove an analogue of Miyachi's theorem for the semi-direct product K ⋉ ℝ. This allows us to solve the sharpness problems in the theorem of Cowling-Price and in the L − L analogue of Morgan theorem for any compact extension of ℝ. These upshots are proved using the representations theory and the Plancherel formula for the group Fourier transform. [ABSTRACT FROM AUTHOR]

Let G be an exponential solvable Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for the homogeneous space $${\fancyscript{M}=G/H}$$ and any deformation of Γ, the deformed discrete subgroup may utterly destroy its proper discontinuous action on $${\fancyscript{M}}$$ as H is not compact (except the case when it is trivial). To understand this specific issue, we provide an explicit description of the parameter and the deformation spaces of any abelian discrete Γ acting properly discontinuously and fixed point freely on G/ H for an arbitrary H of a threadlike nilpotent Lie group G. The topological features of deformations, such as the local rigidity and the stability are also discussed. Whenever the Clifford-Klein form Γ\ G/ H in question is assumed to be compact, these spaces are cutely determined and unlike the case of Heisenberg groups, the deformation space fails in general to be a Hausdorff space. We show further that this space admits a smooth manifold as its open dense subset. [ABSTRACT FROM AUTHOR]