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A Wiener-Tauberian and a Pompeiu type theorems on the Laguerre hypergroup. (English) Zbl 1157.43002

Let (\(\mathbb{K},\,*_\alpha)\) be the Laguerre hypergroup, where \(\mathbb{K}=[0,\,\infty)\times \mathbb{R}\) and \(*_\alpha\) is the convolution product on this hypergroup defined for two bounded Radon measures \(\mu\) and \(\nu\) on \(\mathbb{K}\) by
\[ \mu*_\alpha\nu(f)=\int_{\mathbb{K}\times\mathbb{K}}T_{(\chi,t)}^{(\alpha)}f(y,s)d\mu(\chi,t)d\nu(y,s), \]
where \(\alpha\) is a fixed nonnegative real number and \(\{T_{\chi,t}^{(\alpha)}\}_{(\chi,t)\in \mathbb{K}}\) are the translation operators on the Laguerre hypergroup, given by
\[ T_{(\chi,t)}^{(\alpha)}f(y,s)=\frac {\alpha}{\pi}\int_0^1\int_0^{2\pi}f(((\chi,t),(y,s))_{\theta,r})r(1-r^2)^{\alpha-1}d\theta \,dr, \]
for \(\alpha>0\), and
\[ T_{(\chi,t)}^{(0)}f(y,s)=\frac 1{2\pi}\int_0^{2\pi}f(((\chi,t),(y,s))_{\theta,1})\,d\theta, \]
where
\[ ((\chi,t),(y,s))_{\theta,r}=((\chi^2+y^2+2\chi yr\cos \theta)^{1/2}, t+s+\chi yr\sin \theta). \]
This paper deals with a Wiener-Tauberian theorem on the hypergroup and a Pompeiu type theorem is also established by the Wiener-Tauberian theorem, the corresponding results on the Heisenberg group and associated with the Dunkl operator on the real line can be found in the previous works of M. Agranovsky, C. Berenstein, D.-C. Chang and D. Pascuas [J. Anal. Math. 63, 131–173 (1994; Zbl 0808.43002)]; A. El Garna and B. Selmi [Integral Transforms Spec. Funct. 16, No. 4, 301–314 (2005; Zbl 1069.33015)], respectively. In addition, some applications of the Pompeiu type theorem are given.

MSC:

43A62 Harmonic analysis on hypergroups
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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