Inversion Formulas for the Dunkl Intertwining Operator and Its Dual on Spaces of Functions and Distributions

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 give some applications.Comment: This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

The positivity of the transmutation operators associated to the Cherednik operators for the root system BC2BC_2

We consider the transmutation operators Vk, tVk and V W k , tV W k associated respectively with the Cherednik operators and the Heckman-Opdam theory attached to the root system BC2, called also in [8, 9, 10] the trigonometric Dunkl intertwining operators, and their dual. In this paper we prove that the operators Vk, tVk and VWk , tVWk are positivity preserving and allows positive integral representations. In particular we deduce that the Opdam-Cherednik and the Heckman-Opdam kernels are positive definite

Bochner-Hecke Theorems for the Weinstein Transform and Application

MSC 2010: 42B10, 44A15In this paper we prove Bochner-Hecke theorems for the Weinstein transform and we give an application to homogeneous distributions

Spectrum of Functions for the Dunkl Transform on R^d

Mathematics Subject Classification: 42B10In this paper, we establish real Paley-Wiener theorems for the Dunkl transform on R^d. More precisely, we characterize the functions in the Schwartz space S(R^d) and in L^2k(R^d) whose Dunkl transform has bounded, unbounded, convex and nonconvex support

The positivity of the transmutation operators associated to the Cherednik operators for the root system BC2BC_2

We consider the transmutation operators Vk, tVk and V W k , tV W k associated respectively with the Cherednik operators and the Heckman-Opdam theory attached to the root system BC2, called also in [8, 9, 10] the trigonometric Dunkl intertwining operators, and their dual. In this paper we prove that the operators Vk, tVk and VWk , tVWk are positivity preserving and allows positive integral representations. In particular we deduce that the Opdam-Cherednik and the Heckman-Opdam kernels are positive definite

Biorthogonal multiresolution analyses and decompositions of Sobolev spaces

The object of this paper is to construct extension operators in the Sobolev spaces Hk(]−∞,0]) and Hk([0,+∞[)(k≥0). Then we use these extensions to get biorthogonal wavelet bases in Hk(ℝ). We also give a construction in L2([−1,1]) to see how to obtain boundaries functions