A Paley–Wiener theorem for the inverse spherical transform

Pasquale, Angela

doi:10.2140/pjm.2000.193.143

Cited by 15 publications

(12 citation statements)

References 10 publications

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“…More precisely, 20) whereh n,t * (x) = h n,t * (−x). To see this, first compute f pointwise by the Fourier inversion formula…”

Section: Wwwmn-journalcommentioning

confidence: 99%

Paley‐Wiener estimates for the Heisenberg group

Führ

2010

Mathematische Nachrichten

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The Paley-Wiener space P W (G) on a stratified Lie group G is defined via the spectral decomposition of the associated sub-Laplacian. In this paper, we show that functions in P W (H), where H denotes the Heisenberg group, extend to an entire function on the complexification H C , satisfying a growth estimate of exponential order two. We also show that a converse, characterizing elements of P W (H) only in terms of pointwise growth behaviour of the entire extension, is not available.

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“…More precisely, 20) whereh n,t * (x) = h n,t * (−x). To see this, first compute f pointwise by the Fourier inversion formula…”

Section: Wwwmn-journalcommentioning

confidence: 99%

Paley‐Wiener estimates for the Heisenberg group

Führ

2010

Mathematische Nachrichten

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“…These spaces have previously been considered in [16] and [1]. Let G/K be a Riemannian globally symmetric space of Helgason's noncompact type [6, Chap.…”

Section: The Fourier Transform On a Symmetric Space Of Noncompact Typementioning

confidence: 99%

A Reconstruction Theorem for Riemannian Symmetric Spaces of Noncompact Type

Stenzel

2009

J Fourier Anal Appl

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Let f be a rapidly decreasing radial function on a Riemannian symmetric space of noncompact type whose spherical Fourier transform has compact support. We prove a reconstruction theorem which recovers f from the values of an integral operator applied to f on a discrete subset. When G/K is of the complex type we prove a sampling formula recovering f from its own values on a discrete subset. We give explicit results for three dimensional hyperbolic space.

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“…Then f does not extend to an entire function on a C , due to the poles of the spherical function ϕ λ (exp(H)). There is, however, a description of the Paley-Wiener space PW(K\G/K) as functions having an explicit meromorphic extension and satisfying some exponential growth conditions for the rank 1 and the complex cases, see [8] for details.…”

Section: The Inverse Fourier Transformmentioning

confidence: 99%

“…The paper [8] provides an answer to the above question for the spherical transform on Schwartz functions in the rank one and complex cases. The characterisation is in analogy with the classical Paley-Wiener theorem given in terms of meromorphic extensions and growth conditions.…”

Section: Introductionmentioning

confidence: 99%