POSITIVITY OF THE INTERTWINING OPERATOR AND HARMONIC ANALYSIS ASSOCIATED WITH THE JACOBI-DUNKL OPERATOR ON ${\mathbb R}$
POSITIVITY OF THE INTERTWINING OPERATOR AND HARMONIC ANALYSIS ASSOCIATED WITH THE JACOBI-DUNKL OPERATOR ON ${\mathbb R}$
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DOI：
10.1142/s0219530503000247
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发表时间：
2003-10
影响因子：
2.2
通讯作者：
F. Chouchane;M. Mili;K. Trimeche
中科院分区：
文献类型：
--
作者：
F. Chouchane;M. Mili;K. Trimeche
考虑${\mathbb R}$上的微分-差分算子Λα，β，$\alpha > -\frac{1}{2}$，$\beta\in {\mathbb R}$.该算子的本征函数在零点等于1，称为雅可比-邓克尔核。我们给出了这个函数的一个拉普拉斯积分表示，并证明了对于$\alpha\ge\beta\ge -\frac{1}{2}$，$\alpha\ne -\frac{1}{2}$，这个积分表示的核是正的。这个结果使我们能够证明Jacobi-Dunkl交织算子及其对偶是正的。接下来我们研究了与算子Λα，β相关的调和分析（Jacobi-Dunkl变换，Jacobi-Dunkl平移算子，Jacobi-Dunkl卷积积，Paley-Wiener和Plancherel定理.）。
We consider a differential-difference operator Λα,β, $\alpha > -\frac{1}{2}$, $\beta\in {\mathbb R}$ on ${\mathbb R}$. The eigenfunction of this operator equal to 1 at zero is called the Jacobi–Dunkl kernel. We give a Laplace integral representation for this function and we prove that for $\alpha\ge\beta\ge -\frac{1}{2}$, $\alpha\ne -\frac{1}{2}$, the kernel of this integral representation is positive. This result permits us to prove that the Jacobi–Dunkl intertwining operator and its dual are positive. Next we study the harmonic analysis associated with the operator Λα,β (Jacobi–Dunkl transform, Jacobi–Dunkl translation operators, Jacobi–Dunkl convolution product, Paley–Wiener and Plancherel theorems…).