Related papers: Positivity of Dunkl's intertwining operator
Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to…
These are the lecture notes of a series of lectures on Dunkl operators. We discuss the underlying algebraic structure of the degenerate double affine Hecke algebra, intertwiners and shift operators. We apply this to Macdonald theory. We…
A systematic study of the representation theory of double affine Hecke algebras and related harmonic analysis is started in this paper. Continuing the previous papers we use the technique of intertwining operators to create Macdonald…
Commuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special…
We construct a large family of commutative algebras of partial differential operators invariant under rotations. These algebras are isomorphic extensions of the algebras of ordinary differential operators introduced by Grunbaum and Yakimov…
For a family of weight functions, $h_\kappa$, invariant under a finite reflection group on $\RR^d$, analysis related to the Dunkl transform is carried out for the weighted $L^p$ spaces. Making use of the generalized translation operator and…
Let $T_{b}$ be the Dunkl operator for the reflection group $G=\mathbb{Z}/2\mathbb{Z}$, and $D_{b}:=|x|^{b}\,T_{b}\,|x|^{-b}$. We compute explicitly the unitary one-parameter group $e^{tD_{b}}$ generated by $D_{b}$. We obtain two…
This paper grew out of the author's work on arXiv:2504.18460. Differential operators in the sense of Grothendieck acting between modules over a commutative ring can be interpreted as torsion elements in the bimodule of all operators with…
We introduce a family of differential-reflection operators $\Lambda_{A, \varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is a Strum-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ For…
In this paper, we establish an integral expression for the Dunkl kernel in the context of Dihedral group of an arbitrary order by using the results in \cite{M-Y-Vk} where a construction of the Dunkl intertwining operator for a large set of…
We prove that the spherical mean value of the Dunkl-type generalized translation operator $\tau^y$ is a positive $L^p$-bounded generalized translation operator $T^t$. As application, we prove the Young inequality for a convolution defined…
In this article, we first improve the scalar maximal theorem for the Dunkl maximal operator by giving some precisions on the behavior of the constants of this theorem for a general reflection group. Next we complete the vector-valued…
In this paper we prove inversion formulas for the Dunkl intertwining operator $V_k$ and for its dual ${}^tV_k$ and we deduce the expression of the representing distributions of the inverse operators $V_k^{-1}$ and ${}^tV_k^{-1}$, and we…
We introduce braided Dunkl operators that are acting on a q-polynomial algebra and q-commute. Generalizing the approach of Etingof and Ginzburg, we explain the q-commutation phenomenon by constructing braided Cherednik algebras for which…
The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry…
The rational Dunkl operators are commuting differential-reflection operators on the Euclidean space $\RR^d$ associated with a root system. The aim of the paper is to study local boundary behaviour of generalized harmonic functions…
In this paper we continue to investigate a certain class of Hankel-like positive definite kernels using their associated orthogonal polynomials. The main result of this paper is about the structure of this kind of kernels.
We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer-Fock…
We analyze reflection positive representations in terms of positive Hankel operators. This is motivated by the fact that positive Hankel operators are described in terms of their Carleson measures, whereas the compatibility condition…
We introduce a deformation of the Fourier transform on $\mathbb{R}^N$ arising from a representation-theoretic construction associated with $\widetilde{SL}(2,\mathbb{R}) \times O(N)$ that still admits an underlying degree-one operator…