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

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

© Hindawi Publishing Corp. BIORTHOGONAL MULTIRESOLUTION ANALYSES AND DECOMPOSITIONS OF SOBOLEV SPACES

Abstract. The object of this paper is to construct extension operators in the Sobolev spaces H k (]−∞,0]) and H k ([0,+∞[) (k ≥ 0). Then we use these extensions to get biorthogonal wavelet bases in H k (R). We also give a construction in L 2 ([−1,1]) to see how to obtain boundaries functions. 2000 Mathematics Subject Classification. 41A58, 42C15