Real Paley-Wiener type theorems for the Dunkl transform on
{\mathcal{S}}'(\mathbb{R}^{d}) |
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Authors: | Sihem Ayadi Slaim Ben Farah |
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Institution: | (1) Faculty of Sciences of Monastir, Department of Mathematics, 5019 Monastir, Tunisia |
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Abstract: | We prove real Paley-Wiener type theorems for the Dunkl transform ℱ
D
on the space
of tempered distributions. Let T∈S′(ℝ
d
) and Δ
κ
the Dunkl Laplacian operator. First, we establish that the support of ℱ
D
(T) is included in the Euclidean ball
, M>0, if and only if for all R>M we have lim
n→+∞
R
−2n
Δ
κ
n
T=0 in S′(ℝ
d
). Second, we prove that the support of ℱ
D
(T) is included in ℝ
d
∖B(0,M), M>0, if and only if for all R<M, we have lim
n→+∞
R
2n
ℱ
D
−1(‖y‖−2n
ℱ
D
(T))=0 in S′(ℝ
d
). Finally, we study real Paley-Wiener theorems associated with
-slowly increasing function.
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Keywords: | Tempered distribution Dunkl transform Paley-Wiener theorems |
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