In this paper, we study a few versions of the uncertainty principle for windowed Opdam–Cherednik transform. particular, establish orthonormal sequences, Donoho–Stark's principle, Benedicks‐type Heisenberg‐type and local inequality We also obtain using ‐entropy

In this note we give a new proof of the Macdonald-Mehta-Opdam integral identity for finite Coxeter groups. This identity was conjectured by Macdonald and proved by Opdam in [O1, O2] using the theory of multivariable Bessel functions, but in non-crystallographic cases the proof relied on a computer calculation by F. Garvan. Our proof is somewhat more elementary (in particular, it does not use mu...

In this paper, we study a class of pseudodifferential operators known as time-frequency localization operators, which depend on symbol $\varsigma$ and two windows functions $g_1$ $g_2$. We first present some basic properties the windowed Opdam-Cherednik transform. Then, use modulation spaces associated with transform appropriate classes for symbols windows, boundedness compactness spaces. Final...

A series expansion for Heckman-Opdam hypergeometric functions φλ is obtained for all λ ∈ a∗ C . As a consequence, estimates for φλ away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the no...

We prove certain duality properties and present recurrence relations for a four-parameter family of self-dual Koornwinder-Macdonald polynomials. The recurrence relations are used to verify Macdonald's normalization conjectures for these polynomials. 1. Introduction In a to date unpublished but well-known manuscript, Macdonald introduced certain families of multivariable orthogonal polynomials a...

We study the rational Cherednik algebra Hc of type G(r, 1, n) by means of the Dunkl-Opdam operators introduced in [DuOp] and the generalized Jack polynomials introduced in [Gri]. Our main result is a characterization of the set of parameters c for which Hce+Hc 6= Hc, where e+ is the symmetrizing idempotent for G(r, 1, n). Such parameters are called aspherical. The proof is parallel to that in [...

We construct and study some finite dimensional modules for rational Cherednik algebras for the groups G(r, p, n) by using intertwining operators and a commutative family of operators introduced by Dunkl and Opdam. The coinvariant ring and an analog of the ring constructed by Gordon in the course of proving Haiman’s conjectures on diagonal coinvariants are special cases. We study a certain hyper...

The rational Cherednik algebra H is a certain algebra of differential-reflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S of W corresponds to a standard module M(λ) for H. This paper deals with the infinite family G(r, 1, n) of complex reflection groups; our goal is to study the standard modules using a co...