In this note, a new proof for the positivity of Dunkl’s intertwining operator in the crystallographic case is given. It is based on an asymptotic relationship between the Opdam-Cherednik kernel and the Dunkl kernel as recently observed by M. de Jeu, and on positivity results of S. Sahi for the Heckman-Opdam polynomials and their nonsymmetric counterparts. 2000 AMS Subject Classification: 33C52,...

Consider a root system of type BC1 on the real line R with general positive multiplicities. The Cherednik-Opdam transform defines a unitary operator from an L-space on R to a L-space of C-valued functions on R with the Harish-Chandra measure |c(λ)|dλ. By introducing a weight function of the form cosh(t) tanh t on R we find an orthogonal basis for the L-space on R consisting of even and odd func...

‎In this paper‎, ‎using a generalized translation operator‎, ‎we obtain a generalization of Younis Theorem 5.2 in [3] for the Cherednik-Opdam transform for functions satisfying the $(delta,gamma,p)$-Cherednik-Opdam Lipschitz condition in the space‎ ‎$L^{p}_{alpha,beta}(mathbb{R})$.

The aim of this paper is to establish a few qualitative uncertainty principles for the windowed Opdam–Cherednik transform on weighted modulation spaces associated with transform. In particular, we obtain Cowling–Price's, Hardy's and Morgan's spaces. proofs results are based versions Phragmén–Lindlöf type result several complex variables properties Gaussian kernel Jacobi–Cherednik operator.

In this paper, we introduce the Hausdorff operator associated with Opdam--Cherednik transform and study boundedness of in various Lebesgue spaces. particular, prove spaces, grand quasi-Banach spaces that are transform. Also, give necessary sufficient conditions for these

Abstract In this paper, we discuss a generalization to the Cherednik–Opdam integral operator an abstract space of Boehmians. We introduce sets Boehmians and establish delta sequences certain class convolution products. Then prove that extended is linear, bijective continuous with respect convergence generalized spaces Moreover, derive embeddings properties theory. obtain inversion formula provi...

We continue a program generalizing classical results from the analysis on symmetric cones to Dunkl setting for root systems of type A A ...

We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner–Pollaczek type polynomials. This is done by computing the Cherednik–Opdam transform of the non-symmetric Jack polynomials multiplied by the exponential function.

We conjecture exact estimates for the Weyl-invariant Opdam–Cherednik hypergeometric functions. prove root system $A_n$ and all rank 1 cases. provide other evidence that might be true in general.

In this paper, we study several weighted norm inequalities for the Opdam–Cherednik transform. We establish different versions of Heisenberg–Pauli–Weyl inequality particular, give an extension using weights with exponents and present a variation that incorporates [Formula: see text]-norms Also, prove version Hardy–Littlewood Finally, other variations such as Nash-type Clarkson-type