Generalized Spectral Projections on Symmetric Spaces of Noncompact Type: Paley–Wiener Theorems

Bray, William O.

doi:10.1006/jfan.1996.0009

Cited by 21 publications

(20 citation statements)

References 17 publications

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“…[Koo75] and [Bra96] in the rank-one case. Not only a Paley-Wiener theorem for the inverse spherical transform has not been considered so far, but also the various estimates required in its proof for the rank-one case are different from those considered in other Paley-Wiener type theorems.…”

Section: Introductionmentioning

confidence: 99%

A Paley–Wiener theorem for the inverse spherical transform

Pasquale¹

2000

Pacific J. Math.

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Section: Introductionmentioning

confidence: 99%

A Paley–Wiener theorem for the inverse spherical transform

Pasquale¹

2000

Pacific J. Math.

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“…There is a refinement of this theorem, due to Strichartz [24] and Bray [4], in terms of the spectral projections f * Φ λ where Φ λ are spherical functions on G/K. For each irreducible unitary representation δ of K with a unique Mfixed vector, there are functions Y δ on K/M which play the role of spherical harmonics.…”

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confidence: 99%

Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces

Thangavelu

2002

Colloq. Math.

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Abstract. Let G be a semisimple Lie group with Iwasawa decomposition G = KAN . Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis {Y δ,j :, λ ∈ C, be the Helgason Fourier transform. Let h t be the heat kernel associated to the Laplace-Beltrami operator and let Q δ (iλ + ) be the Kostant polynomials. We establish the following version of Hardy's theorem for the Helgason Fourier transform: Let f be a function on G/K which satisfies |f (ka r )| ≤ Ch t (r). Further assume that for every δ and j the functions2 for λ ∈ R. Then f is a constant multiple of the heat kernel h t .

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“…Thus (f (λ)Y 0 , Y δj ) is Q δ (iλ + ρ) times the Jacobi transform of f δ,j (a). For all these and more about Jacobi transform we refer to Anker et al [2], Bray [7] and [39]. Let p δ t be the heat kernel associated to the operator ∆ δ which is characterised by…”

Section: Corollary 34mentioning

confidence: 99%

On Theorems of Hardy, Gelfand–Shilov and Beurling for Semisimple Groups

Thangavelu¹

2004

Publ. Res. Inst. Math. Sci.

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In this paper we prove a strong version of Hardy's theorem for the group Fourier transform on semisimple Lie groups which characterises the Fourier transforms of all functions satisfying Hardy type conditions. In the particular case of SL(2, IR) we characterise all such functions and conjecture that the same is true for all rank one semisimple groups. We also establish an analogue of a theorem of Gelfand and Shilov in the context of semisimple groups. A version of Beurling's theorem which assumes a Cowling-Price condition on the function is also proved. We show that these results yield most of the earlier results as corollaries. §1. Introduction and the Main ResultsThe aim of this paper is to establish a couple of uncertainty principles such as the theorems of Hardy, Gelfand-Shilov and Beurling for semisimple

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Generalized Spectral Projections on Symmetric Spaces of Noncompact Type: Paley–Wiener Theorems

Cited by 21 publications

References 17 publications

A Paley–Wiener theorem for the inverse spherical transform

A Paley–Wiener theorem for the inverse spherical transform

Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces

On Theorems of Hardy, Gelfand–Shilov and Beurling for Semisimple Groups

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