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Table Of Integrals Series And Products

Real and Complex Analysis

The Theory of Groups and Quantum Mechanics

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Reflection Groups And Coxeter Groups

The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography.

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Mathematics of thermoacoustic tomography

We use the Dunkl intertwining operator V_k and its dual ^tV_k to prove a Paley-Wiener theorem for the Dunkl transform, for functions and distributions, and geometric forms of this theorem. These operators permit also to define and study Dunkl translation operators and Dunkl convolution product.

Paley-Wiener Theorems for the Dunkl Transform and Dunkl Translation Operators

In this paper we prove that the Dunkl intertwining operator Vk defined only on the space of polynomials on can also be extended to an operator on spaces of functions and distributions. We study also its dual tVk and we prove that for each there exists a positive measure vy on the Borel ω-algebra of with support in such that for all function f continuous on and with compact support, we have

The dunkel intertwining operator on spaces of functions and distributions and integral representation of its dual

Inversion of the Radon Transform on the Laguerre Hypergroup by using Generalized Wavelets

Ranges and inversion formulas for spherical mean operator and its dual

We consider a singular differential-difference operator Λ on the real line which includes, as particular case, the Dunkl operator associated with the reflection group Z2 on R. We exhibit a Laplace integral representation for the eigenfunctions of the operator Λ. From this representation, we construct a pair of integral transforms which turn out to be transmutation operators of Λ into the first derivative operator d/dx. We exploit these transmutation operators to develop a new commutative harmonic analysis on the real line corresponding to the operator Λ. In particular, we establish a Paley–Wiener theorem and a Plancherel theorem for the Fourier transform associated to Λ.

Khalifa Trimèche

Papers

Paley-Wiener Theorems for the Dunkl Transform and Dunkl Translation Operators

The dunkel intertwining operator on spaces of functions and distributions and integral representation of its dual

Inversion of the Radon Transform on the Laguerre Hypergroup by using Generalized Wavelets

Ranges and inversion formulas for spherical mean operator and its dual

Transmutation operators and paley–wiener theorem associated with a singular differential-difference operator on the real line