Remarks on the inverse Cherednik-Opdam transform on the real line

Uncertainty principles for the Opdam-Cherednik transform on modulation spaces

In this paper, we establish the Cowling--Price's, Hardy's and Morgan's uncertainty principles for the Opdam-Cherednik transform on modulation spaces associated with this transform. The proofs of the theorems are based on the properties of the heat kernel associated with the Jacobi-Cherednik operator and the versions of the Phragm{é}n-Lindl{ö}f type result for the modulation spaces.

调制空间上与加窗 Opdam-Cherednik 变换相关的局部化算子

Localization operators associated with the windowed Opdam-Cherednik transform on modulation spaces

本文以加窗 Opdam-Cherednik 变换为基础，在相应调制空间框架下研究依赖符号与双窗函数的时频局部化算子，建立其有界性、紧性及 Schatten-von Neumann 类归属。

Opdam-Cherednik核是正测度的拉普拉斯变换

The Opdam-Cherednik kernel is the Laplace transform of a positive measure

本文证明Opdam-Cherednik核可表示为支撑在其参数Weyl群轨道凸包上的正测度的拉普拉斯变换，从而三角Dunkl交织算子保持正性。

Opdam's hypergeometric functions: product formula and convolution structure in dimension 1

Let $G_λ^{(α,β)}$ be the eigenfunctions of the Dunkl-Cherednik operator $T^{(α,β)}$ on $\mathbb{R}$. In this paper we express the product $G_λ^{(α,β)}(x)G_λ^{(α,β)}(y)$ as an integral in terms of $G_λ^{(α,β)}(z)$ with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called rational limit, we recover the product formula of M. Rösler for the Dunkl kernel. We then define and study a convolution structure associated to $G_λ^{(α,β)}$.

Non-symmetric Jacobi and Wilson type polynomials

Consider a root system of type $BC_1$ on the real line $\mathbb R$ with general positive multiplicities. The Cherednik-Opdam transform defines a unitary operator from an $L^2$-space on $\mathbb R$ to a $L^2$-space of $\mathbb C^2$-valued functions on $\mathbb R^+$ with the Harish-Chandra measure $|c(\lam)|^{-2}d\lam$. By introducing a weight function of the form $\cosh^{-\sig}(t)\tanh^{2k} t$ on $\mathbb R$ we find an orthogonal basis for the $L^2$-space on $\mathbb R$ consisting of even and odd functions expressed in terms of the Jacobi polynomials (for each fixed $\sig$ and $k$). We find a R…

Poisson核与Hilbert变换在三角Heckman-Opdam多项式类型$A_1$中的应用

Poisson Kernels and Hilbert Transforms for Trigonometric Heckman-Opdam Polynomials of type $A_1$

本文研究了类型$A_1$的三角Heckman-Opdam多项式，推导了Poisson核的显式表达式，并发展了与这些多项式相关的分数积分理论，同时定义了广义Hilbert变换并证明其在$L^p$空间上的有界性。