This paper is concerned with energy properties of the wave equation associated to the Dunkl-Cherednik Laplacian. We establish the conservation of the total energy, the strict equipartition of energy under suitable assumptions and the asymptotic equipartition in the general case. Mathematics Subject Index 2000: Primary 35L05 ; Secondary 22E30, 33C67, 35L65, 58J45

Purpose Paley's and Hardy's inequality are proved on a Hardy-type space for the Fourier–Dunkl expansions based complete orthonormal system of Dunkl kernels generalizing classical exponential defining Fourier series. Design/methodology/approach Although difficulties related to settings, techniques used by K. Sato were still efficient in this case establish inequalities which have expected simila...

In this paper, we proved a weighted Hardy–Rellich inequality for Dunkl operators based on the spherical h-harmonic decomposition theory of operators. Moreover, obtained explicit constant inequalities, which is optimal in some cases. Our results extend known inequalities.

A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an orthonormal system which is complete in L2((−1, 1), |x|2α+1 dx). This orthonormal system is a generalization of the classical exponential system defining Fourier series.

For a class of weight functions invariant under reflection groups on the unit ball, a family of orthogonal polynomials is defined via a Rodrigues type formula using the Dunkl operators. Their properties and their relation with several other bases are explored.