Cubature Formulae Associated with the Dunkl Laplacian

Abstract

In this paper, we study the integration of functions of the form

$$u=\sum_{j=0}^{m-1}u_{j}(x) \Phi_{j}(|x|^{2}),$$

where (u _j ) are in ${\mathcal{C}^{1}(\overline{B(r)})\cap\mathcal{C}^{2}(B(r))}$ and harmonic in the open ball B(r) centered at the origin and with radius r > 0, with respect to the Dunkl Laplacian Δ _k and ${\{\Phi_{0},\ldots,\Phi_{m-1}\}}$ is a given system of linearly independent integrable functions on [0, r ²]. In particular, we construct cubature formulae having highest order of precision with respect to the class of k-polyharmonic functions of degree m, i.e. ${\Delta_{k}^{m}u=0,m\in\mathbb{N}\setminus\{0\}}$ and we give an extension of the Pizzetti formula type for functions in ${\mathcal{C}^{2m-1}(\overline{B(r)}) \cap\mathcal{C}^{2m}(B(r))}$ and k-polyharmonic of order m.