Spectrum of Functions and Distributions for the Jacobi-Dunkl Transform on $${\mathbb{R}}$$

Mejjaoli, Hatem

doi:10.1007/s00009-012-0206-4

Cited by 12 publications

(10 citation statements)

References 11 publications

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“…We refer to [1], [10], [11], [12], [14] [15], [16], [17], [34], [35], [36], [41], [45], [47], [63] for more details. Finally, we mention results by Pesenson in a similar vein; see [49], [50], [51], [52], [53].…”

Section: Literature Reviewmentioning

confidence: 99%

“…The results and approach mentioned above were in [72] and [45] generalized to the Dunkl transform, where we recall that the Dunkl transform, [37], [42], is a deformation of the Fourier transform that specializes to the Fourier transform if all the so-called root multiplicities are zero. This also means that we believe the results in [72] and [45] are valid only for real valued polynomials.…”

Section: Literature Reviewmentioning

confidence: 99%

“…This also means that we believe the results in [72] and [45] are valid only for real valued polynomials.…”

Section: Literature Reviewmentioning

confidence: 99%

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Real Paley–Wiener theorems and local spectral radius formulas

Andersen¹,

Jeu²

2010

Trans. Amer. Math. Soc.

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Abstract. We systematically develop real Paley-Wiener theory for the Fourier transform on R d for Schwartz functions, L p -functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley-Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use for other transforms of Fourier type as well. An explanation is also given as to why the easily applied classical Paley-Wiener theorems are unlikely to be able to yield information about the support of a function or distribution which is more precise than giving its convex hull, whereas real Paley-Wiener theorems can be used to reconstruct the support precisely, albeit at the cost of combinatorial complexity. We indicate a possible application of real Paley-Wiener theory to partial differential equations in this vein, and furthermore we give evidence that a number of real PaleyWiener results can be expected to have an interpretation as local spectral radius formulas. A comprehensive overview of the literature on real PaleyWiener theory is included.

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Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

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Real Paley–Wiener theorems and local spectral radius formulas

Andersen¹,

Jeu²

2010

Trans. Amer. Math. Soc.

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“…The Plancherel's and inversion theorems are also established for this transform. Very recently, many authors have been investigating the behavior of the Dunkl transform with respect to several problems already studied for the Fourier transform; for instance, uncertainty [4], Besov spaces [5], real Paley-Wiener theorems [6], generalized Sonine-type integral transforms [7], heat equation [8], maximal function [10], and so on.…”

Section: Introductionmentioning

confidence: 99%

A new version of the real Paley-Wiener theorems

Mejjaoli

2017

Filomat

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“…Next the analogue of this theorem was established and improved for many other integral transforms, for examples (cf. [1,6,14,[16][17][18]23]). The second subject concerning spectral theorems is the study of tempered distributions with spectral gaps.…”

Section: Introductionmentioning

confidence: 99%

Spectral Theorems Associated to the Dunkl Operators

Mejjaoli

2016

Korean Journal of Mathematics

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Abstract. In this paper, we characterize the support for the Dunkl transform on the generalized Lebesgue spaces via the Dunkl resolvent function. The behavior of the sequence of L p k −norms of iterated Dunkl potentials is studied depending on the support of their Dunkl transform. We systematically develop real Paley-Wiener theory for the Dunkl transform on R d for distributions, in an elementary treatment based on the inversion theorem. Next, we improve the Roe's theorem associated to the Dunkl operators.

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Spectrum of Functions and Distributions for the Jacobi-Dunkl Transform on $${\mathbb{R}}$$

Cited by 12 publications

References 11 publications

Real Paley–Wiener theorems and local spectral radius formulas

Real Paley–Wiener theorems and local spectral radius formulas

A new version of the real Paley-Wiener theorems

Spectral Theorems Associated to the Dunkl Operators

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