Sinc-type functions on a class of nilpotent Lie groups

Oussa, Vignon

doi:10.1515/apam-2013-0028

Cited by 3 publications

(8 citation statements)

References 15 publications

(27 reference statements)

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“…where S is a finite subset of Z n−2d and each (Λ − κ j ) ∩ E • is a set of positive Lebesgue measure on R n−2d . For λ ∈ Σ, we define the map λ → u λ on Σ such that (7) u…”

Section: Overview Of Main Resultsmentioning

confidence: 99%

“…The specific definition of bandlimited spaces by the Plancherel transform used in [3], was taken from [5], Chapter 6, where a very precise characterization of sampling spaces over the Heisenberg group was provided. Moreover, sampling spaces using a similar definition of bandlimitation were studied in [8] and [7] for a class of nilpotent Lie groups which contains the Heisenberg Lie groups. This class of groups was first introduced by the author in [8].…”

Section: Introductionmentioning

confidence: 99%

“…We organize this paper as follows. The second section deals with some preliminary results which can be found in [8,7,2]. In the third section, we introduce a natural notion of bandlimitation for the class of groups considered, and we state the main results (Theorem 7 and Theorem 8) of the paper.…”

Section: Introductionmentioning

confidence: 99%

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Sampling and interpolation on some nilpotent Lie groups

Oussa

2014

Forum Mathematicum

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Let N be a non-commutative, simply connected, connected, two-step nilpotent Lie group with Lie algebra n such that,j≤d is a non-vanishing homogeneous polynomial in the unknowns Z 1 , · · · , Z n−2d where {Z 1 , · · · , Z n−2d } is a basis for the center of the Lie algebra. Using well-known facts from time-frequency analysis, we provide some precise sufficient conditions for the existence of sampling spaces with the interpolation property, with respect to some discrete subset of N . The result obtained in this work can be seen as a direct application of time-frequency analysis to the theory of nilpotent Lie groups. Several explicit examples are computed. This work is a generalization of recent results obtained for the Heisenberg group by Currey, and Mayeli in [3].

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Section: Overview Of Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sampling and interpolation on some nilpotent Lie groups

Oussa

2014

Forum Mathematicum

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“…Firstly, we observe that if N = R d then Γ can be taken to be an integer lattice, and the Hilbert space of functions vanishing outside the cube - It is shown in [9,6] that there exist subspaces of L 2 (N) which are sampling subspaces with respect to Γ. We have also established in [19,18,17] the existence of sampling spaces defined over a class of simply connected, connected nilpotent Lie groups which satisfy the following conditions: N is a step-two nilpotent Lie group with Lie algebra n of dimension n such that n = a ⊕ b ⊕ c where [a, b] ⊆ c, a, b are commutative Lie algebras, a = R-span {X 1 , X 2 , · · · , X d } , b = R-span {Y 1 , Y 2 , · · · , Y d } , c = R-span {Z 1 , Z 2 , · · · , Z n-2d } (d ≥ 1, n > 2d) and…”

Section: Introductionmentioning

confidence: 94%

Regular Sampling on Metabelian Nilpotent Lie Groups: The Multiplicity-Free Case

Oussa

2017

Frames and Other Bases in Abstract and Function Spaces

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Let N be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra n having rational structure constants. We assume that N = P M, M is commutative, and for all λ ∈ n * in general position the subalgebra p = log(P) is a polarization ideal subordinated to λ (p is a maximal ideal satisfying [p, p] ⊆ ker λ for all λ in general position and p is necessarily commutative.) Under these assumptions, we prove that there exists a discrete uniform subgroup Γ ⊂ N such that L 2 (N) admits band-limited spaces with respect to the group Fourier transform which are sampling spaces with respect to Γ. We also provide explicit sufficient conditions which are easily checked for the existence of sampling spaces. Sufficient conditions for sampling spaces which enjoy the interpolation property are also given. Our result bears a striking resemblance with the well-known Whittaker-Kotel'nikov-Shannon sampling theorem.

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“…For each λ ∈ Λ the corresponding irreducible representation π λ is realized as acting on L 2 R 3 as follows (see [13] for more details)…”

Section: Preliminariesmentioning

confidence: 99%

Decompositions of generalized wavelet representations

Currey¹,

Mayeli²,

Oussa³

2014

Contemporary Mathematics

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Abstract. Let N be a simply connected, connected nilpotent Lie group which admits a uniform subgroup Γ. Let α be an automorphism of N defined by α (exp X) = exp AX. We assume that the linear action of A is diagonalizable and we do not assume that N is commutative. Let W be a unitary wavelet representation of the semi-direct product group. We obtain a decomposition of W into a direct integral of unitary representations. Moreover, we provide an explicit unitary operator intertwining the representations, a precise description of the representations occurring, the measure used in the direct integral decomposition and the support of the measure. We also study the irreducibility of the fiber representations occurring in the direct integral decomposition in various settings. We prove that in the case where A is an expansive automorphism then the decomposition of W is in fact a direct integral of unitary irreducible representations each occurring with infinite multiplicities if and only if N is non-commutative. This work naturally extends results obtained by H. Lim, J. Packer and K. Taylor who obtained a direct integral decomposition of W in the case where N is commutative and the matrix A is expansive, i.e. all eigenvalues have absolute values larger than one.

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Sinc-type functions on a class of nilpotent Lie groups

Abstract: Let N be a simply connected, connected nilpotent Lie group with the following assumptions. Its Lie algebra n is an n-dimensional vector space over the reals. Moreover, n = z ⊕ b ⊕ a, z is the center of n, z = RZ n−2d ⊕ RZ n−2d−

Cited by 3 publications

References 15 publications

Sampling and interpolation on some nilpotent Lie groups

Sampling and interpolation on some nilpotent Lie groups

Regular Sampling on Metabelian Nilpotent Lie Groups: The Multiplicity-Free Case

Decompositions of generalized wavelet representations

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