Harmonic analysis associated with the multivariate Laguerre function (Q1635732)

Harmonic analysis on the multi-dimensional Laguerre hypergroup \(\mathbb K=[0,\infty)^n\times \mathbb R\) is studied. For a given multi-index \(\alpha=(\alpha_1,\ldots,\alpha_n)\in(0,\infty)^n\) the hypergroup structure on \(\mathbb K\) is introduced via a product formula for the functions \(\Psi^\alpha_{m,\lambda}(x,t)=e^{i\lambda t}\mathcal L^\alpha_m(|\lambda|x^2)\), \(m\in\mathbb N^n\), \(\lambda\in\mathbb R\), where \(\mathcal L^\alpha_m\) denotes a multi-dimensional Laguerre function of degree \(m\) and order \(\alpha\), and \(x^2=(x_1^2,\ldots,x_n^2)\) for \(x=(x_1,\ldots,x_n)\in[0,\infty)^n\). For an associated Fourier-Laguerre transform a Plancherel theorem and an inversion formula are established and a Paley-Wiener-type theorem is proved.